Finite Math Examples

$x + 3 y - 4 z - w = 8$ , $4 x + y + 2 z + 5 w = 12$ , $- 2 x - y + z - 2 w = - 9$ , $x - y - 3 z - 3 w = - 4$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

Subtract $3 y$ from both sides of the equation.

$x - 4 z - w = 8 - 3 y$

$4 x + y + 2 z + 5 w = 12$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

Step 1.2

Add $4 z$ to both sides of the equation.

$x - w = 8 - 3 y + 4 z$

$4 x + y + 2 z + 5 w = 12$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

Step 1.3

Add $w$ to both sides of the equation.

$x = 8 - 3 y + 4 z + w$

$4 x + y + 2 z + 5 w = 12$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

$x = 8 - 3 y + 4 z + w$

$4 x + y + 2 z + 5 w = 12$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

Step 2

Replace all occurrences of $x$ with $8 - 3 y + 4 z + w$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $4 x + y + 2 z + 5 w = 12$ with $8 - 3 y + 4 z + w$ .

$4 (8 - 3 y + 4 z + w) + y + 2 z + 5 w = 12$

$x = 8 - 3 y + 4 z + w$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $4 (8 - 3 y + 4 z + w) + y + 2 z + 5 w$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$4 \cdot 8 + 4 (- 3 y) + 4 (4 z) + 4 w + y + 2 z + 5 w = 12$

$x = 8 - 3 y + 4 z + w$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $4$ by $8$ .

$32 + 4 (- 3 y) + 4 (4 z) + 4 w + y + 2 z + 5 w = 12$

$x = 8 - 3 y + 4 z + w$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

Step 2.2.1.1.2.2

Multiply $- 3$ by $4$ .

$32 - 12 y + 4 (4 z) + 4 w + y + 2 z + 5 w = 12$

$x = 8 - 3 y + 4 z + w$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

Step 2.2.1.1.2.3

Multiply $4$ by $4$ .

$32 - 12 y + 16 z + 4 w + y + 2 z + 5 w = 12$

$x = 8 - 3 y + 4 z + w$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

$32 - 12 y + 16 z + 4 w + y + 2 z + 5 w = 12$

$x = 8 - 3 y + 4 z + w$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

$32 - 12 y + 16 z + 4 w + y + 2 z + 5 w = 12$

$x = 8 - 3 y + 4 z + w$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $- 12 y$ and $y$ .

$32 - 11 y + 16 z + 4 w + 2 z + 5 w = 12$

$x = 8 - 3 y + 4 z + w$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

Step 2.2.1.2.2

Add $16 z$ and $2 z$ .

$32 - 11 y + 4 w + 18 z + 5 w = 12$

$x = 8 - 3 y + 4 z + w$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

Step 2.2.1.2.3

Add $4 w$ and $5 w$ .

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$- 2 x - y + z - 2 w = - 9$

$x - y - 3 z - 3 w = - 4$

Step 2.3

Replace all occurrences of $x$ in $- 2 x - y + z - 2 w = - 9$ with $8 - 3 y + 4 z + w$ .

$- 2 (8 - 3 y + 4 z + w) - y + z - 2 w = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$x - y - 3 z - 3 w = - 4$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- 2 (8 - 3 y + 4 z + w) - y + z - 2 w$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$- 2 \cdot 8 - 2 (- 3 y) - 2 (4 z) - 2 w - y + z - 2 w = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$x - y - 3 z - 3 w = - 4$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 2$ by $8$ .

$- 16 - 2 (- 3 y) - 2 (4 z) - 2 w - y + z - 2 w = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$x - y - 3 z - 3 w = - 4$

Step 2.4.1.1.2.2

Multiply $- 3$ by $- 2$ .

$- 16 + 6 y - 2 (4 z) - 2 w - y + z - 2 w = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$x - y - 3 z - 3 w = - 4$

Step 2.4.1.1.2.3

Multiply $4$ by $- 2$ .

$- 16 + 6 y - 8 z - 2 w - y + z - 2 w = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$x - y - 3 z - 3 w = - 4$

$- 16 + 6 y - 8 z - 2 w - y + z - 2 w = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$x - y - 3 z - 3 w = - 4$

$- 16 + 6 y - 8 z - 2 w - y + z - 2 w = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$x - y - 3 z - 3 w = - 4$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $y$ from $6 y$ .

$- 16 + 5 y - 8 z - 2 w + z - 2 w = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$x - y - 3 z - 3 w = - 4$

Step 2.4.1.2.2

Add $- 8 z$ and $z$ .

$- 16 + 5 y - 2 w - 7 z - 2 w = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$x - y - 3 z - 3 w = - 4$

Step 2.4.1.2.3

Subtract $2 w$ from $- 2 w$ .

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$x - y - 3 z - 3 w = - 4$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$x - y - 3 z - 3 w = - 4$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$x - y - 3 z - 3 w = - 4$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$x - y - 3 z - 3 w = - 4$

Step 2.5

Replace all occurrences of $x$ in $x - y - 3 z - 3 w = - 4$ with $8 - 3 y + 4 z + w$ .

$(8 - 3 y + 4 z + w) - y - 3 z - 3 w = - 4$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 2.6

Simplify the left side.

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Step 2.6.1

Simplify $(8 - 3 y + 4 z + w) - y - 3 z - 3 w$ .

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Step 2.6.1.1

Subtract $y$ from $- 3 y$ .

$8 - 4 y + 4 z + w - 3 z - 3 w = - 4$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 2.6.1.2

Subtract $3 z$ from $4 z$ .

$8 - 4 y + w + z - 3 w = - 4$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 2.6.1.3

Subtract $3 w$ from $w$ .

$8 - 4 y - 2 w + z = - 4$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$8 - 4 y - 2 w + z = - 4$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$8 - 4 y - 2 w + z = - 4$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$8 - 4 y - 2 w + z = - 4$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 3

Move all terms not containing $z$ to the right side of the equation.

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Step 3.1

Subtract $8$ from both sides of the equation.

$- 4 y - 2 w + z = - 4 - 8$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 3.2

Add $4 y$ to both sides of the equation.

$- 2 w + z = - 4 - 8 + 4 y$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 3.3

Add $2 w$ to both sides of the equation.

$z = - 4 - 8 + 4 y + 2 w$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 3.4

Subtract $8$ from $- 4$ .

$z = - 12 + 4 y + 2 w$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$z = - 12 + 4 y + 2 w$

$- 16 + 5 y - 4 w - 7 z = - 9$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 4

Replace all occurrences of $z$ with $- 12 + 4 y + 2 w$ in each equation.

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Step 4.1

Replace all occurrences of $z$ in $- 16 + 5 y - 4 w - 7 z = - 9$ with $- 12 + 4 y + 2 w$ .

$- 16 + 5 y - 4 w - 7 (- 12 + 4 y + 2 w) = - 9$

$z = - 12 + 4 y + 2 w$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 16 + 5 y - 4 w - 7 (- 12 + 4 y + 2 w)$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$- 16 + 5 y - 4 w - 7 \cdot - 12 - 7 (4 y) - 7 (2 w) = - 9$

$z = - 12 + 4 y + 2 w$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 4.2.1.1.2

Simplify.

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Step 4.2.1.1.2.1

Multiply $- 7$ by $- 12$ .

$- 16 + 5 y - 4 w + 84 - 7 (4 y) - 7 (2 w) = - 9$

$z = - 12 + 4 y + 2 w$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 4.2.1.1.2.2

Multiply $4$ by $- 7$ .

$- 16 + 5 y - 4 w + 84 - 28 y - 7 (2 w) = - 9$

$z = - 12 + 4 y + 2 w$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 4.2.1.1.2.3

Multiply $2$ by $- 7$ .

$- 16 + 5 y - 4 w + 84 - 28 y - 14 w = - 9$

$z = - 12 + 4 y + 2 w$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$- 16 + 5 y - 4 w + 84 - 28 y - 14 w = - 9$

$z = - 12 + 4 y + 2 w$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$- 16 + 5 y - 4 w + 84 - 28 y - 14 w = - 9$

$z = - 12 + 4 y + 2 w$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Add $- 16$ and $84$ .

$5 y - 4 w + 68 - 28 y - 14 w = - 9$

$z = - 12 + 4 y + 2 w$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 4.2.1.2.2

Subtract $28 y$ from $5 y$ .

$- 23 y - 4 w + 68 - 14 w = - 9$

$z = - 12 + 4 y + 2 w$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 4.2.1.2.3

Subtract $14 w$ from $- 4 w$ .

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$32 - 11 y + 9 w + 18 z = 12$

$x = 8 - 3 y + 4 z + w$

Step 4.3

Replace all occurrences of $z$ in $32 - 11 y + 9 w + 18 z = 12$ with $- 12 + 4 y + 2 w$ .

$32 - 11 y + 9 w + 18 (- 12 + 4 y + 2 w) = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y + 4 z + w$

Step 4.4

Simplify the left side.

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Step 4.4.1

Simplify $32 - 11 y + 9 w + 18 (- 12 + 4 y + 2 w)$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$32 - 11 y + 9 w + 18 \cdot - 12 + 18 (4 y) + 18 (2 w) = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y + 4 z + w$

Step 4.4.1.1.2

Simplify.

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Step 4.4.1.1.2.1

Multiply $18$ by $- 12$ .

$32 - 11 y + 9 w - 216 + 18 (4 y) + 18 (2 w) = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y + 4 z + w$

Step 4.4.1.1.2.2

Multiply $4$ by $18$ .

$32 - 11 y + 9 w - 216 + 72 y + 18 (2 w) = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y + 4 z + w$

Step 4.4.1.1.2.3

Multiply $2$ by $18$ .

$32 - 11 y + 9 w - 216 + 72 y + 36 w = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y + 4 z + w$

$32 - 11 y + 9 w - 216 + 72 y + 36 w = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y + 4 z + w$

$32 - 11 y + 9 w - 216 + 72 y + 36 w = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y + 4 z + w$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $216$ from $32$ .

$- 11 y + 9 w - 184 + 72 y + 36 w = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y + 4 z + w$

Step 4.4.1.2.2

Add $- 11 y$ and $72 y$ .

$61 y + 9 w - 184 + 36 w = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y + 4 z + w$

Step 4.4.1.2.3

Add $9 w$ and $36 w$ .

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y + 4 z + w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y + 4 z + w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y + 4 z + w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y + 4 z + w$

Step 4.5

Replace all occurrences of $z$ in $x = 8 - 3 y + 4 z + w$ with $- 12 + 4 y + 2 w$ .

$x = 8 - 3 y + 4 (- 12 + 4 y + 2 w) + w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 4.6

Simplify the right side.

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Step 4.6.1

Simplify $8 - 3 y + 4 (- 12 + 4 y + 2 w) + w$ .

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Step 4.6.1.1

Simplify each term.

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Step 4.6.1.1.1

Apply the distributive property.

$x = 8 - 3 y + 4 \cdot - 12 + 4 (4 y) + 4 (2 w) + w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 4.6.1.1.2

Simplify.

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Step 4.6.1.1.2.1

Multiply $4$ by $- 12$ .

$x = 8 - 3 y - 48 + 4 (4 y) + 4 (2 w) + w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 4.6.1.1.2.2

Multiply $4$ by $4$ .

$x = 8 - 3 y - 48 + 16 y + 4 (2 w) + w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 4.6.1.1.2.3

Multiply $2$ by $4$ .

$x = 8 - 3 y - 48 + 16 y + 8 w + w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y - 48 + 16 y + 8 w + w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 8 - 3 y - 48 + 16 y + 8 w + w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 4.6.1.2

Simplify by adding terms.

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Step 4.6.1.2.1

Subtract $48$ from $8$ .

$x = - 3 y - 40 + 16 y + 8 w + w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 4.6.1.2.2

Add $- 3 y$ and $16 y$ .

$x = 13 y - 40 + 8 w + w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 4.6.1.2.3

Add $8 w$ and $w$ .

$x = 13 y - 40 + 9 w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 13 y - 40 + 9 w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 13 y - 40 + 9 w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 13 y - 40 + 9 w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = 13 y - 40 + 9 w$

$61 y + 45 w - 184 = 12$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 5

Solve for $y$ in $61 y + 45 w - 184 = 12$ .

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Step 5.1

Move all terms not containing $y$ to the right side of the equation.

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Step 5.1.1

Subtract $45 w$ from both sides of the equation.

$61 y - 184 = 12 - 45 w$

$x = 13 y - 40 + 9 w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 5.1.2

Add $184$ to both sides of the equation.

$61 y = 12 - 45 w + 184$

$x = 13 y - 40 + 9 w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 5.1.3

Add $12$ and $184$ .

$61 y = - 45 w + 196$

$x = 13 y - 40 + 9 w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$61 y = - 45 w + 196$

$x = 13 y - 40 + 9 w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 5.2

Divide each term in $61 y = - 45 w + 196$ by $61$ and simplify.

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Step 5.2.1

Divide each term in $61 y = - 45 w + 196$ by $61$ .

$\frac{61 y}{61} = \frac{- 45 w}{61} + \frac{196}{61}$

$x = 13 y - 40 + 9 w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $61$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{61 y}{61} = \frac{- 45 w}{61} + \frac{196}{61}$

$x = 13 y - 40 + 9 w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 45 w}{61} + \frac{196}{61}$

$x = 13 y - 40 + 9 w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$y = \frac{- 45 w}{61} + \frac{196}{61}$

$x = 13 y - 40 + 9 w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$y = \frac{- 45 w}{61} + \frac{196}{61}$

$x = 13 y - 40 + 9 w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

$y = - \frac{45 w}{61} + \frac{196}{61}$

$x = 13 y - 40 + 9 w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$x = 13 y - 40 + 9 w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$x = 13 y - 40 + 9 w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$x = 13 y - 40 + 9 w$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6

Replace all occurrences of $y$ with $- \frac{45 w}{61} + \frac{196}{61}$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = 13 y - 40 + 9 w$ with $- \frac{45 w}{61} + \frac{196}{61}$ .

$x = 13 (- \frac{45 w}{61} + \frac{196}{61}) - 40 + 9 w$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $13 (- \frac{45 w}{61} + \frac{196}{61}) - 40 + 9 w$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Apply the distributive property.

$x = 13 (- \frac{45 w}{61}) + 13 (\frac{196}{61}) - 40 + 9 w$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.1.2

Multiply $13 (- \frac{45 w}{61})$ .

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Step 6.2.1.1.2.1

Multiply $- 1$ by $13$ .

$x = - 13 \frac{45 w}{61} + 13 (\frac{196}{61}) - 40 + 9 w$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.1.2.2

Combine $- 13$ and $\frac{45 w}{61}$ .

$x = \frac{- 13 (45 w)}{61} + 13 (\frac{196}{61}) - 40 + 9 w$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.1.2.3

Multiply $45$ by $- 13$ .

$x = \frac{- 585 w}{61} + 13 (\frac{196}{61}) - 40 + 9 w$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = \frac{- 585 w}{61} + 13 (\frac{196}{61}) - 40 + 9 w$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.1.3

Multiply $13 (\frac{196}{61})$ .

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Step 6.2.1.1.3.1

Combine $13$ and $\frac{196}{61}$ .

$x = \frac{- 585 w}{61} + \frac{13 \cdot 196}{61} - 40 + 9 w$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.1.3.2

Multiply $13$ by $196$ .

$x = \frac{- 585 w}{61} + \frac{2548}{61} - 40 + 9 w$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = \frac{- 585 w}{61} + \frac{2548}{61} - 40 + 9 w$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.1.4

Move the negative in front of the fraction.

$x = - \frac{585 w}{61} + \frac{2548}{61} - 40 + 9 w$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = - \frac{585 w}{61} + \frac{2548}{61} - 40 + 9 w$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.2

To write $9 w$ as a fraction with a common denominator, multiply by $\frac{61}{61}$ .

$x = - \frac{585 w}{61} + 9 w \cdot \frac{61}{61} + \frac{2548}{61} - 40$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.3

Combine $9 w$ and $\frac{61}{61}$ .

$x = - \frac{585 w}{61} + \frac{9 w \cdot 61}{61} + \frac{2548}{61} - 40$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.4

Combine the numerators over the common denominator.

$x = \frac{- 585 w + 9 w \cdot 61}{61} + \frac{2548}{61} - 40$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.5

Combine the numerators over the common denominator.

$x = - 40 + \frac{- 585 w + 9 w \cdot 61 + 2548}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.6

Multiply $61$ by $9$ .

$x = - 40 + \frac{- 585 w + 549 w + 2548}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.7

Add $- 585 w$ and $549 w$ .

$x = - 40 + \frac{- 36 w + 2548}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.8

Factor $4$ out of $- 36 w + 2548$ .

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Step 6.2.1.8.1

Factor $4$ out of $- 36 w$ .

$x = - 40 + \frac{4 (- 9 w) + 2548}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.8.2

Factor $4$ out of $2548$ .

$x = - 40 + \frac{4 (- 9 w) + 4 (637)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.8.3

Factor $4$ out of $4 (- 9 w) + 4 (637)$ .

$x = - 40 + \frac{4 (- 9 w + 637)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = - 40 + \frac{4 (- 9 w + 637)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.9

To write $- 40$ as a fraction with a common denominator, multiply by $\frac{61}{61}$ .

$x = - 40 \cdot \frac{61}{61} + \frac{4 (- 9 w + 637)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.10

Simplify terms.

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Step 6.2.1.10.1

Combine $- 40$ and $\frac{61}{61}$ .

$x = \frac{- 40 \cdot 61}{61} + \frac{4 (- 9 w + 637)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.10.2

Combine the numerators over the common denominator.

$x = \frac{- 40 \cdot 61 + 4 (- 9 w + 637)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = \frac{- 40 \cdot 61 + 4 (- 9 w + 637)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.11

Simplify the numerator.

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Step 6.2.1.11.1

Factor $4$ out of $- 40 \cdot 61 + 4 (- 9 w + 637)$ .

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Step 6.2.1.11.1.1

Factor $4$ out of $- 40 \cdot 61$ .

$x = \frac{4 (- 10 \cdot 61) + 4 (- 9 w + 637)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.11.1.2

Factor $4$ out of $4 (- 10 \cdot 61) + 4 (- 9 w + 637)$ .

$x = \frac{4 (- 10 \cdot 61 - 9 w + 637)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = \frac{4 (- 10 \cdot 61 - 9 w + 637)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.11.2

Multiply $- 10$ by $61$ .

$x = \frac{4 (- 610 - 9 w + 637)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.11.3

Add $- 610$ and $637$ .

$x = \frac{4 (- 9 w + 27)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.11.4

Factor $9$ out of $- 9 w + 27$ .

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Step 6.2.1.11.4.1

Factor $9$ out of $- 9 w$ .

$x = \frac{4 (9 (- w) + 27)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.11.4.2

Factor $9$ out of $27$ .

$x = \frac{4 (9 (- w) + 9 (3))}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.11.4.3

Factor $9$ out of $9 (- w) + 9 (3)$ .

$x = \frac{4 (9 (- w + 3))}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = \frac{4 \cdot (9 (- w + 3))}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.11.5

Multiply $4$ by $9$ .

$x = \frac{36 (- w + 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = \frac{36 (- w + 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.12

Simplify with factoring out.

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Step 6.2.1.12.1

Factor $- 1$ out of $- w$ .

$x = \frac{36 (- (w) + 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.12.2

Rewrite $3$ as $- 1 (- 3)$ .

$x = \frac{36 (- (w) - 1 \cdot - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.12.3

Factor $- 1$ out of $- (w) - 1 (- 3)$ .

$x = \frac{36 (- (w - 3))}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.12.4

Simplify the expression.

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Step 6.2.1.12.4.1

Rewrite $- (w - 3)$ as $- 1 (w - 3)$ .

$x = \frac{36 (- 1 (w - 3))}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.2.1.12.4.2

Move the negative in front of the fraction.

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$- 23 y - 18 w + 68 = - 9$

$z = - 12 + 4 y + 2 w$

Step 6.3

Replace all occurrences of $y$ in $- 23 y - 18 w + 68 = - 9$ with $- \frac{45 w}{61} + \frac{196}{61}$ .

$- 23 (- \frac{45 w}{61} + \frac{196}{61}) - 18 w + 68 = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4

Simplify the left side.

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Step 6.4.1

Simplify $- 23 (- \frac{45 w}{61} + \frac{196}{61}) - 18 w + 68$ .

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Step 6.4.1.1

Simplify each term.

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Step 6.4.1.1.1

Apply the distributive property.

$- 23 (- \frac{45 w}{61}) - 23 (\frac{196}{61}) - 18 w + 68 = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.1.2

Multiply $- 23 (- \frac{45 w}{61})$ .

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Step 6.4.1.1.2.1

Multiply $- 1$ by $- 23$ .

$23 (\frac{45 w}{61}) - 23 (\frac{196}{61}) - 18 w + 68 = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.1.2.2

Combine $23$ and $\frac{45 w}{61}$ .

$\frac{23 (45 w)}{61} - 23 (\frac{196}{61}) - 18 w + 68 = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.1.2.3

Multiply $45$ by $23$ .

$\frac{1035 w}{61} - 23 (\frac{196}{61}) - 18 w + 68 = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

$\frac{1035 w}{61} - 23 (\frac{196}{61}) - 18 w + 68 = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.1.3

Multiply $- 23 (\frac{196}{61})$ .

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Step 6.4.1.1.3.1

Combine $- 23$ and $\frac{196}{61}$ .

$\frac{1035 w}{61} + \frac{- 23 \cdot 196}{61} - 18 w + 68 = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.1.3.2

Multiply $- 23$ by $196$ .

$\frac{1035 w}{61} + \frac{- 4508}{61} - 18 w + 68 = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

$\frac{1035 w}{61} + \frac{- 4508}{61} - 18 w + 68 = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.1.4

Move the negative in front of the fraction.

$\frac{1035 w}{61} - \frac{4508}{61} - 18 w + 68 = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

$\frac{1035 w}{61} - \frac{4508}{61} - 18 w + 68 = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.2

To write $- 18 w$ as a fraction with a common denominator, multiply by $\frac{61}{61}$ .

$\frac{1035 w}{61} - 18 w \cdot \frac{61}{61} - \frac{4508}{61} + 68 = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.3

Combine $- 18 w$ and $\frac{61}{61}$ .

$\frac{1035 w}{61} + \frac{- 18 w \cdot 61}{61} - \frac{4508}{61} + 68 = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.4

Combine the numerators over the common denominator.

$\frac{1035 w - 18 w \cdot 61}{61} - \frac{4508}{61} + 68 = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.5

Combine the numerators over the common denominator.

$68 + \frac{1035 w - 18 w \cdot 61 - 4508}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.6

Multiply $61$ by $- 18$ .

$68 + \frac{1035 w - 1098 w - 4508}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.7

Subtract $1098 w$ from $1035 w$ .

$68 + \frac{- 63 w - 4508}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.8

Factor $7$ out of $- 63 w - 4508$ .

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Step 6.4.1.8.1

Factor $7$ out of $- 63 w$ .

$68 + \frac{7 (- 9 w) - 4508}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.8.2

Factor $7$ out of $- 4508$ .

$68 + \frac{7 (- 9 w) + 7 (- 644)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.8.3

Factor $7$ out of $7 (- 9 w) + 7 (- 644)$ .

$68 + \frac{7 (- 9 w - 644)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

$68 + \frac{7 (- 9 w - 644)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.9

To write $68$ as a fraction with a common denominator, multiply by $\frac{61}{61}$ .

$68 \cdot \frac{61}{61} + \frac{7 (- 9 w - 644)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.10

Simplify terms.

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Step 6.4.1.10.1

Combine $68$ and $\frac{61}{61}$ .

$\frac{68 \cdot 61}{61} + \frac{7 (- 9 w - 644)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.10.2

Combine the numerators over the common denominator.

$\frac{68 \cdot 61 + 7 (- 9 w - 644)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

$\frac{68 \cdot 61 + 7 (- 9 w - 644)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.11

Simplify the numerator.

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Step 6.4.1.11.1

Multiply $68$ by $61$ .

$\frac{4148 + 7 (- 9 w - 644)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.11.2

Apply the distributive property.

$\frac{4148 + 7 (- 9 w) + 7 \cdot - 644}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.11.3

Multiply $- 9$ by $7$ .

$\frac{4148 - 63 w + 7 \cdot - 644}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.11.4

Multiply $7$ by $- 644$ .

$\frac{4148 - 63 w - 4508}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.11.5

Subtract $4508$ from $4148$ .

$\frac{- 63 w - 360}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.11.6

Factor $9$ out of $- 63 w - 360$ .

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Step 6.4.1.11.6.1

Factor $9$ out of $- 63 w$ .

$\frac{9 (- 7 w) - 360}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.11.6.2

Factor $9$ out of $- 360$ .

$\frac{9 (- 7 w) + 9 (- 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.11.6.3

Factor $9$ out of $9 (- 7 w) + 9 (- 40)$ .

$\frac{9 (- 7 w - 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

$\frac{9 (- 7 w - 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

$\frac{9 (- 7 w - 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.12

Simplify with factoring out.

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Step 6.4.1.12.1

Factor $- 1$ out of $- 7 w$ .

$\frac{9 (- (7 w) - 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.12.2

Rewrite $- 40$ as $- 1 (40)$ .

$\frac{9 (- (7 w) - 1 \cdot 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.12.3

Factor $- 1$ out of $- (7 w) - 1 (40)$ .

$\frac{9 (- (7 w + 40))}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.12.4

Simplify the expression.

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Step 6.4.1.12.4.1

Rewrite $- (7 w + 40)$ as $- 1 (7 w + 40)$ .

$\frac{9 (- 1 (7 w + 40))}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.4.1.12.4.2

Move the negative in front of the fraction.

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + 4 y + 2 w$

Step 6.5

Replace all occurrences of $y$ in $z = - 12 + 4 y + 2 w$ with $- \frac{45 w}{61} + \frac{196}{61}$ .

$z = - 12 + 4 (- \frac{45 w}{61} + \frac{196}{61}) + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6

Simplify the right side.

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Step 6.6.1

Simplify $- 12 + 4 (- \frac{45 w}{61} + \frac{196}{61}) + 2 w$ .

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Step 6.6.1.1

Simplify each term.

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Step 6.6.1.1.1

Apply the distributive property.

$z = - 12 + 4 (- \frac{45 w}{61}) + 4 (\frac{196}{61}) + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.1.2

Multiply $4 (- \frac{45 w}{61})$ .

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Step 6.6.1.1.2.1

Multiply $- 1$ by $4$ .

$z = - 12 - 4 \frac{45 w}{61} + 4 (\frac{196}{61}) + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.1.2.2

Combine $- 4$ and $\frac{45 w}{61}$ .

$z = - 12 + \frac{- 4 (45 w)}{61} + 4 (\frac{196}{61}) + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.1.2.3

Multiply $45$ by $- 4$ .

$z = - 12 + \frac{- 180 w}{61} + 4 (\frac{196}{61}) + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + \frac{- 180 w}{61} + 4 (\frac{196}{61}) + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.1.3

Multiply $4 (\frac{196}{61})$ .

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Step 6.6.1.1.3.1

Combine $4$ and $\frac{196}{61}$ .

$z = - 12 + \frac{- 180 w}{61} + \frac{4 \cdot 196}{61} + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.1.3.2

Multiply $4$ by $196$ .

$z = - 12 + \frac{- 180 w}{61} + \frac{784}{61} + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 + \frac{- 180 w}{61} + \frac{784}{61} + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.1.4

Move the negative in front of the fraction.

$z = - 12 - \frac{180 w}{61} + \frac{784}{61} + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 12 - \frac{180 w}{61} + \frac{784}{61} + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.2

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{61}{61}$ .

$z = - \frac{180 w}{61} - 12 \cdot \frac{61}{61} + \frac{784}{61} + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.3

Combine $- 12$ and $\frac{61}{61}$ .

$z = - \frac{180 w}{61} + \frac{- 12 \cdot 61}{61} + \frac{784}{61} + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.4

Combine the numerators over the common denominator.

$z = - \frac{180 w}{61} + \frac{- 12 \cdot 61 + 784}{61} + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.5

Simplify the numerator.

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Step 6.6.1.5.1

Multiply $- 12$ by $61$ .

$z = - \frac{180 w}{61} + \frac{- 732 + 784}{61} + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.5.2

Add $- 732$ and $784$ .

$z = - \frac{180 w}{61} + \frac{52}{61} + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - \frac{180 w}{61} + \frac{52}{61} + 2 w$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.6

To write $2 w$ as a fraction with a common denominator, multiply by $\frac{61}{61}$ .

$z = - \frac{180 w}{61} + 2 w \cdot \frac{61}{61} + \frac{52}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.7

Combine $2 w$ and $\frac{61}{61}$ .

$z = - \frac{180 w}{61} + \frac{2 w \cdot 61}{61} + \frac{52}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.8

Combine the numerators over the common denominator.

$z = \frac{- 180 w + 2 w \cdot 61}{61} + \frac{52}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.9

Combine the numerators over the common denominator.

$z = \frac{- 180 w + 2 w \cdot 61 + 52}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.10

Multiply $61$ by $2$ .

$z = \frac{- 180 w + 122 w + 52}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.11

Add $- 180 w$ and $122 w$ .

$z = \frac{- 58 w + 52}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.12

Factor $2$ out of $- 58 w + 52$ .

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Step 6.6.1.12.1

Factor $2$ out of $- 58 w$ .

$z = \frac{2 (- 29 w) + 52}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.12.2

Factor $2$ out of $52$ .

$z = \frac{2 (- 29 w) + 2 (26)}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.12.3

Factor $2$ out of $2 (- 29 w) + 2 (26)$ .

$z = \frac{2 (- 29 w + 26)}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = \frac{2 (- 29 w + 26)}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.13

Factor $- 1$ out of $- 29 w$ .

$z = \frac{2 (- (29 w) + 26)}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.14

Rewrite $26$ as $- 1 (- 26)$ .

$z = \frac{2 (- (29 w) - 1 \cdot - 26)}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.15

Factor $- 1$ out of $- (29 w) - 1 (- 26)$ .

$z = \frac{2 (- (29 w - 26))}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.16

Simplify the expression.

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Step 6.6.1.16.1

Rewrite $- (29 w - 26)$ as $- 1 (29 w - 26)$ .

$z = \frac{2 (- 1 (29 w - 26))}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 6.6.1.16.2

Move the negative in front of the fraction.

$z = - \frac{2 (29 w - 26)}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - \frac{2 (29 w - 26)}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - \frac{2 (29 w - 26)}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - \frac{2 (29 w - 26)}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - \frac{2 (29 w - 26)}{61}$

$- \frac{9 (7 w + 40)}{61} = - 9$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7

Solve for $w$ in $- \frac{9 (7 w + 40)}{61} = - 9$ .

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Step 7.1

Multiply both sides of the equation by $- \frac{61}{9}$ .

$- \frac{61}{9} \cdot (- \frac{9 (7 w + 40)}{61}) = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2

Simplify both sides of the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Simplify $- \frac{61}{9} (- \frac{9 (7 w + 40)}{61})$ .

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Step 7.2.1.1.1

Cancel the common factor of $61$ .

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Step 7.2.1.1.1.1

Move the leading negative in $- \frac{61}{9}$ into the numerator.

$\frac{- 61}{9} \cdot (- \frac{9 (7 w + 40)}{61}) = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2.1.1.1.2

Move the leading negative in $- \frac{9 (7 w + 40)}{61}$ into the numerator.

$\frac{- 61}{9} \cdot \frac{- 9 (7 w + 40)}{61} = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2.1.1.1.3

Factor $61$ out of $- 61$ .

$\frac{61 (- 1)}{9} \cdot \frac{- 9 (7 w + 40)}{61} = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2.1.1.1.4

Cancel the common factor.

$\frac{61 \cdot - 1}{9} \cdot \frac{- 9 (7 w + 40)}{61} = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2.1.1.1.5

Rewrite the expression.

$\frac{- 1}{9} \cdot (- 9 (7 w + 40)) = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$\frac{- 1}{9} \cdot (- 9 (7 w + 40)) = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2.1.1.2

Cancel the common factor of $9$ .

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Step 7.2.1.1.2.1

Factor $9$ out of $- 9 (7 w + 40)$ .

$\frac{- 1}{9} \cdot (9 (- (7 w + 40))) = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2.1.1.2.2

Cancel the common factor.

$\frac{- 1}{9} \cdot (9 (- (7 w + 40))) = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2.1.1.2.3

Rewrite the expression.

$7 w + 40 = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$7 w + 40 = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2.1.1.3

Multiply.

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Step 7.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 (7 w + 40) = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2.1.1.3.2

Multiply $7 w + 40$ by $1$ .

$7 w + 40 = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$7 w + 40 = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$7 w + 40 = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$7 w + 40 = - \frac{61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $- \frac{61}{9} \cdot - 9$ .

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Step 7.2.2.1.1

Cancel the common factor of $9$ .

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Step 7.2.2.1.1.1

Move the leading negative in $- \frac{61}{9}$ into the numerator.

$7 w + 40 = \frac{- 61}{9} \cdot - 9$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2.2.1.1.2

Factor $9$ out of $- 9$ .

$7 w + 40 = \frac{- 61}{9} \cdot (9 (- 1))$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2.2.1.1.3

Cancel the common factor.

$7 w + 40 = \frac{- 61}{9} \cdot (9 \cdot - 1)$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2.2.1.1.4

Rewrite the expression.

$7 w + 40 = - 61 \cdot - 1$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$7 w + 40 = - 61 \cdot - 1$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.2.2.1.2

Multiply $- 61$ by $- 1$ .

$7 w + 40 = 61$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$7 w + 40 = 61$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$7 w + 40 = 61$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$7 w + 40 = 61$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.3

Move all terms not containing $w$ to the right side of the equation.

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Step 7.3.1

Subtract $40$ from both sides of the equation.

$7 w = 61 - 40$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.3.2

Subtract $40$ from $61$ .

$7 w = 21$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$7 w = 21$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.4

Divide each term in $7 w = 21$ by $7$ and simplify.

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Step 7.4.1

Divide each term in $7 w = 21$ by $7$ .

$\frac{7 w}{7} = \frac{21}{7}$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Cancel the common factor of $7$ .

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Step 7.4.2.1.1

Cancel the common factor.

$\frac{7 w}{7} = \frac{21}{7}$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.4.2.1.2

Divide $w$ by $1$ .

$w = \frac{21}{7}$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$w = \frac{21}{7}$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$w = \frac{21}{7}$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Divide $21$ by $7$ .

$w = 3$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$w = 3$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$w = 3$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$w = 3$

$z = - \frac{2 (29 w - 26)}{61}$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 8

Replace all occurrences of $w$ with $3$ in each equation.

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Step 8.1

Replace all occurrences of $w$ in $z = - \frac{2 (29 w - 26)}{61}$ with $3$ .

$z = - \frac{2 (29 (3) - 26)}{61}$

$w = 3$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $- \frac{2 (29 (3) - 26)}{61}$ .

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Step 8.2.1.1

Simplify the numerator.

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Step 8.2.1.1.1

Multiply $29$ by $3$ .

$z = - \frac{2 (87 - 26)}{61}$

$w = 3$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 8.2.1.1.2

Subtract $26$ from $87$ .

$z = - \frac{2 \cdot 61}{61}$

$w = 3$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - \frac{2 \cdot 61}{61}$

$w = 3$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 8.2.1.2

Simplify the expression.

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Step 8.2.1.2.1

Multiply $2$ by $61$ .

$z = - \frac{122}{61}$

$w = 3$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 8.2.1.2.2

Divide $122$ by $61$ .

$z = - 1 \cdot 2$

$w = 3$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 8.2.1.2.3

Multiply $- 1$ by $2$ .

$z = - 2$

$w = 3$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 2$

$w = 3$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 2$

$w = 3$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$z = - 2$

$w = 3$

$x = - \frac{36 (w - 3)}{61}$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 8.3

Replace all occurrences of $w$ in $x = - \frac{36 (w - 3)}{61}$ with $3$ .

$x = - \frac{36 ((3) - 3)}{61}$

$z = - 2$

$w = 3$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 8.4

Simplify the right side.

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Step 8.4.1

Simplify $- \frac{36 ((3) - 3)}{61}$ .

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Step 8.4.1.1

Subtract $3$ from $3$ .

$x = - \frac{36 \cdot 0}{61}$

$z = - 2$

$w = 3$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 8.4.1.2

Multiply $36$ by $0$ .

$x = - \frac{0}{61}$

$z = - 2$

$w = 3$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 8.4.1.3

Divide $0$ by $61$ .

$x = - 0$

$z = - 2$

$w = 3$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 8.4.1.4

Multiply $- 1$ by $0$ .

$x = 0$

$z = - 2$

$w = 3$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$x = 0$

$z = - 2$

$w = 3$

$y = - \frac{45 w}{61} + \frac{196}{61}$

$x = 0$

$z = - 2$

$w = 3$

$y = - \frac{45 w}{61} + \frac{196}{61}$

Step 8.5

Replace all occurrences of $w$ in $y = - \frac{45 w}{61} + \frac{196}{61}$ with $3$ .

$y = - \frac{45 (3)}{61} + \frac{196}{61}$

$x = 0$

$z = - 2$

$w = 3$

Step 8.6

Simplify the right side.

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Step 8.6.1

Simplify $- \frac{45 (3)}{61} + \frac{196}{61}$ .

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Step 8.6.1.1

Combine the numerators over the common denominator.

$y = \frac{- 45 \cdot 3 + 196}{61}$

$x = 0$

$z = - 2$

$w = 3$

Step 8.6.1.2

Simplify the expression.

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Step 8.6.1.2.1

Multiply $- 45$ by $3$ .

$y = \frac{- 135 + 196}{61}$

$x = 0$

$z = - 2$

$w = 3$

Step 8.6.1.2.2

Add $- 135$ and $196$ .

$y = \frac{61}{61}$

$x = 0$

$z = - 2$

$w = 3$

Step 8.6.1.2.3

Divide $61$ by $61$ .

$y = 1$

$x = 0$

$z = - 2$

$w = 3$

$y = 1$

$x = 0$

$z = - 2$

$w = 3$

$y = 1$

$x = 0$

$z = - 2$

$w = 3$

$y = 1$

$x = 0$

$z = - 2$

$w = 3$

$y = 1$

$x = 0$

$z = - 2$

$w = 3$

Step 9

List all of the solutions.

$y = 1, x = 0, z = - 2, w = 3$