Finite Math Examples

$x - 2 y + 3 z = 1$ , $y - z + w = - 6$ , $- 3 x + 3 y - 3 z + 4 w = 8$ , $2 y - 3 z + w = 1$

Step 1

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

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Step 1.1

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

$x + 3 z = 1 + 2 y$

$y - z + w = - 6$

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

$2 y - 3 z + w = 1$

Step 1.2

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

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

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

$2 y - 3 z + w = 1$

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

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

$2 y - 3 z + w = 1$

Step 2

Replace all occurrences of $x$ with $1 + 2 y - 3 z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- 3 x + 3 y - 3 z + 4 w = 8$ with $1 + 2 y - 3 z$ .

$- 3 (1 + 2 y - 3 z) + 3 y - 3 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

$2 y - 3 z + w = 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 3 (1 + 2 y - 3 z) + 3 y - 3 z + 4 w$ .

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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.

$- 3 \cdot 1 - 3 (2 y) - 3 (- 3 z) + 3 y - 3 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

$2 y - 3 z + w = 1$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 3$ by $1$ .

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

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

$2 y - 3 z + w = 1$

Step 2.2.1.1.2.2

Multiply $2$ by $- 3$ .

$- 3 - 6 y - 3 (- 3 z) + 3 y - 3 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

$2 y - 3 z + w = 1$

Step 2.2.1.1.2.3

Multiply $- 3$ by $- 3$ .

$- 3 - 6 y + 9 z + 3 y - 3 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

$2 y - 3 z + w = 1$

$- 3 - 6 y + 9 z + 3 y - 3 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

$2 y - 3 z + w = 1$

$- 3 - 6 y + 9 z + 3 y - 3 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

$2 y - 3 z + w = 1$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $- 6 y$ and $3 y$ .

$- 3 - 3 y + 9 z - 3 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

$2 y - 3 z + w = 1$

Step 2.2.1.2.2

Subtract $3 z$ from $9 z$ .

$- 3 - 3 y + 6 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

$2 y - 3 z + w = 1$

$- 3 - 3 y + 6 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

$2 y - 3 z + w = 1$

$- 3 - 3 y + 6 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

$2 y - 3 z + w = 1$

$- 3 - 3 y + 6 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

$2 y - 3 z + w = 1$

$- 3 - 3 y + 6 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$y - z + w = - 6$

$2 y - 3 z + w = 1$

Step 3

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

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Step 3.1

Add $z$ to both sides of the equation.

$y + w = - 6 + z$

$- 3 - 3 y + 6 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

Step 3.2

Subtract $w$ from both sides of the equation.

$y = - 6 + z - w$

$- 3 - 3 y + 6 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

$y = - 6 + z - w$

$- 3 - 3 y + 6 z + 4 w = 8$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

Step 4

Replace all occurrences of $y$ with $- 6 + z - w$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $- 3 - 3 y + 6 z + 4 w = 8$ with $- 6 + z - w$ .

$- 3 - 3 (- 6 + z - w) + 6 z + 4 w = 8$

$y = - 6 + z - w$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 3 - 3 (- 6 + z - w) + 6 z + 4 w$ .

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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.

$- 3 - 3 \cdot - 6 - 3 z - 3 (- w) + 6 z + 4 w = 8$

$y = - 6 + z - w$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

Step 4.2.1.1.2

Simplify.

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Step 4.2.1.1.2.1

Multiply $- 3$ by $- 6$ .

$- 3 + 18 - 3 z - 3 (- w) + 6 z + 4 w = 8$

$y = - 6 + z - w$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

Step 4.2.1.1.2.2

Multiply $- 1$ by $- 3$ .

$- 3 + 18 - 3 z + 3 w + 6 z + 4 w = 8$

$y = - 6 + z - w$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

$- 3 + 18 - 3 z + 3 w + 6 z + 4 w = 8$

$y = - 6 + z - w$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

$- 3 + 18 - 3 z + 3 w + 6 z + 4 w = 8$

$y = - 6 + z - w$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Add $- 3$ and $18$ .

$15 - 3 z + 3 w + 6 z + 4 w = 8$

$y = - 6 + z - w$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

Step 4.2.1.2.2

Add $- 3 z$ and $6 z$ .

$15 + 3 w + 3 z + 4 w = 8$

$y = - 6 + z - w$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

Step 4.2.1.2.3

Add $3 w$ and $4 w$ .

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$x = 1 + 2 y - 3 z$

$2 y - 3 z + w = 1$

Step 4.3

Replace all occurrences of $y$ in $x = 1 + 2 y - 3 z$ with $- 6 + z - w$ .

$x = 1 + 2 (- 6 + z - w) - 3 z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$2 y - 3 z + w = 1$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $1 + 2 (- 6 + z - w) - 3 z$ .

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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.

$x = 1 + 2 \cdot - 6 + 2 z + 2 (- w) - 3 z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$2 y - 3 z + w = 1$

Step 4.4.1.1.2

Simplify.

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Step 4.4.1.1.2.1

Multiply $2$ by $- 6$ .

$x = 1 - 12 + 2 z + 2 (- w) - 3 z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$2 y - 3 z + w = 1$

Step 4.4.1.1.2.2

Multiply $- 1$ by $2$ .

$x = 1 - 12 + 2 z - 2 w - 3 z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$2 y - 3 z + w = 1$

$x = 1 - 12 + 2 z - 2 w - 3 z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$2 y - 3 z + w = 1$

$x = 1 - 12 + 2 z - 2 w - 3 z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$2 y - 3 z + w = 1$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $12$ from $1$ .

$x = - 11 + 2 z - 2 w - 3 z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$2 y - 3 z + w = 1$

Step 4.4.1.2.2

Subtract $3 z$ from $2 z$ .

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$2 y - 3 z + w = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$2 y - 3 z + w = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$2 y - 3 z + w = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$2 y - 3 z + w = 1$

Step 4.5

Replace all occurrences of $y$ in $2 y - 3 z + w = 1$ with $- 6 + z - w$ .

$2 (- 6 + z - w) - 3 z + w = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 4.6

Simplify the left side.

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Step 4.6.1

Simplify $2 (- 6 + z - w) - 3 z + w$ .

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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.

$2 \cdot - 6 + 2 z + 2 (- w) - 3 z + w = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 4.6.1.1.2

Simplify.

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Step 4.6.1.1.2.1

Multiply $2$ by $- 6$ .

$- 12 + 2 z + 2 (- w) - 3 z + w = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 4.6.1.1.2.2

Multiply $- 1$ by $2$ .

$- 12 + 2 z - 2 w - 3 z + w = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$- 12 + 2 z - 2 w - 3 z + w = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$- 12 + 2 z - 2 w - 3 z + w = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 4.6.1.2

Simplify by adding terms.

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Step 4.6.1.2.1

Subtract $3 z$ from $2 z$ .

$- 12 - 2 w - z + w = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 4.6.1.2.2

Add $- 2 w$ and $w$ .

$- 12 - w - z = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$- 12 - w - z = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$- 12 - w - z = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$- 12 - w - z = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$- 12 - w - z = 1$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 5

Solve for $w$ in $- 12 - w - z = 1$ .

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Step 5.1

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

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Step 5.1.1

Add $12$ to both sides of the equation.

$- w - z = 1 + 12$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 5.1.2

Add $z$ to both sides of the equation.

$- w = 1 + 12 + z$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 5.1.3

Add $1$ and $12$ .

$- w = 13 + z$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$- w = 13 + z$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 5.2

Divide each term in $- w = 13 + z$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- w = 13 + z$ by $- 1$ .

$\frac{- w}{- 1} = \frac{13}{- 1} + \frac{z}{- 1}$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{w}{1} = \frac{13}{- 1} + \frac{z}{- 1}$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 5.2.2.2

Divide $w$ by $1$ .

$w = \frac{13}{- 1} + \frac{z}{- 1}$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$w = \frac{13}{- 1} + \frac{z}{- 1}$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Divide $13$ by $- 1$ .

$w = - 13 + \frac{z}{- 1}$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 5.2.3.1.2

Move the negative one from the denominator of $\frac{z}{- 1}$ .

$w = - 13 - 1 \cdot z$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 5.2.3.1.3

Rewrite $- 1 \cdot z$ as $- z$ .

$w = - 13 - z$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$w = - 13 - z$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$w = - 13 - z$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$w = - 13 - z$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$w = - 13 - z$

$x = - 11 - 2 w - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 6

Replace all occurrences of $w$ with $- 13 - z$ in each equation.

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Step 6.1

Replace all occurrences of $w$ in $x = - 11 - 2 w - z$ with $- 13 - z$ .

$x = - 11 - 2 (- 13 - z) - z$

$w = - 13 - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 11 - 2 (- 13 - z) - z$ .

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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.

$x = - 11 - 2 \cdot - 13 - 2 (- z) - z$

$w = - 13 - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 6.2.1.1.2

Multiply $- 2$ by $- 13$ .

$x = - 11 + 26 - 2 (- z) - z$

$w = - 13 - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 6.2.1.1.3

Multiply $- 1$ by $- 2$ .

$x = - 11 + 26 + 2 z - z$

$w = - 13 - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$x = - 11 + 26 + 2 z - z$

$w = - 13 - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 6.2.1.2

Simplify by adding terms.

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Step 6.2.1.2.1

Add $- 11$ and $26$ .

$x = 15 + 2 z - z$

$w = - 13 - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 6.2.1.2.2

Subtract $z$ from $2 z$ .

$x = 15 + z$

$w = - 13 - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$x = 15 + z$

$w = - 13 - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$x = 15 + z$

$w = - 13 - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

$x = 15 + z$

$w = - 13 - z$

$15 + 7 w + 3 z = 8$

$y = - 6 + z - w$

Step 6.3

Replace all occurrences of $w$ in $15 + 7 w + 3 z = 8$ with $- 13 - z$ .

$15 + 7 (- 13 - z) + 3 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = - 6 + z - w$

Step 6.4

Simplify the left side.

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Step 6.4.1

Simplify $15 + 7 (- 13 - z) + 3 z$ .

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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.

$15 + 7 \cdot - 13 + 7 (- z) + 3 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = - 6 + z - w$

Step 6.4.1.1.2

Multiply $7$ by $- 13$ .

$15 - 91 + 7 (- z) + 3 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = - 6 + z - w$

Step 6.4.1.1.3

Multiply $- 1$ by $7$ .

$15 - 91 - 7 z + 3 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = - 6 + z - w$

$15 - 91 - 7 z + 3 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = - 6 + z - w$

Step 6.4.1.2

Simplify by adding terms.

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Step 6.4.1.2.1

Subtract $91$ from $15$ .

$- 76 - 7 z + 3 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = - 6 + z - w$

Step 6.4.1.2.2

Add $- 7 z$ and $3 z$ .

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = - 6 + z - w$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = - 6 + z - w$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = - 6 + z - w$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = - 6 + z - w$

Step 6.5

Replace all occurrences of $w$ in $y = - 6 + z - w$ with $- 13 - z$ .

$y = - 6 + z - (- 13 - z)$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

Step 6.6

Simplify the right side.

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Step 6.6.1

Simplify $- 6 + z - (- 13 - z)$ .

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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.

$y = - 6 + z + 13 + z$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

Step 6.6.1.1.2

Multiply $- 1$ by $- 13$ .

$y = - 6 + z + 13 + z$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

Step 6.6.1.1.3

Multiply $- - z$ .

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Step 6.6.1.1.3.1

Multiply $- 1$ by $- 1$ .

$y = - 6 + z + 13 + 1 z$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

Step 6.6.1.1.3.2

Multiply $z$ by $1$ .

$y = - 6 + z + 13 + z$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = - 6 + z + 13 + z$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = - 6 + z + 13 + z$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

Step 6.6.1.2

Simplify by adding terms.

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Step 6.6.1.2.1

Add $- 6$ and $13$ .

$y = z + 7 + z$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

Step 6.6.1.2.2

Add $z$ and $z$ .

$y = 2 z + 7$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = 2 z + 7$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = 2 z + 7$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = 2 z + 7$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

$y = 2 z + 7$

$- 76 - 4 z = 8$

$x = 15 + z$

$w = - 13 - z$

Step 7

Solve for $z$ in $- 76 - 4 z = 8$ .

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Step 7.1

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

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Step 7.1.1

Add $76$ to both sides of the equation.

$- 4 z = 8 + 76$

$y = 2 z + 7$

$x = 15 + z$

$w = - 13 - z$

Step 7.1.2

Add $8$ and $76$ .

$- 4 z = 84$

$y = 2 z + 7$

$x = 15 + z$

$w = - 13 - z$

$- 4 z = 84$

$y = 2 z + 7$

$x = 15 + z$

$w = - 13 - z$

Step 7.2

Divide each term in $- 4 z = 84$ by $- 4$ and simplify.

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Step 7.2.1

Divide each term in $- 4 z = 84$ by $- 4$ .

$\frac{- 4 z}{- 4} = \frac{84}{- 4}$

$y = 2 z + 7$

$x = 15 + z$

$w = - 13 - z$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $- 4$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{- 4 z}{- 4} = \frac{84}{- 4}$

$y = 2 z + 7$

$x = 15 + z$

$w = - 13 - z$

Step 7.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{84}{- 4}$

$y = 2 z + 7$

$x = 15 + z$

$w = - 13 - z$

$z = \frac{84}{- 4}$

$y = 2 z + 7$

$x = 15 + z$

$w = - 13 - z$

$z = \frac{84}{- 4}$

$y = 2 z + 7$

$x = 15 + z$

$w = - 13 - z$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $84$ by $- 4$ .

$z = - 21$

$y = 2 z + 7$

$x = 15 + z$

$w = - 13 - z$

$z = - 21$

$y = 2 z + 7$

$x = 15 + z$

$w = - 13 - z$

$z = - 21$

$y = 2 z + 7$

$x = 15 + z$

$w = - 13 - z$

$z = - 21$

$y = 2 z + 7$

$x = 15 + z$

$w = - 13 - z$

Step 8

Replace all occurrences of $z$ with $- 21$ in each equation.

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Step 8.1

Replace all occurrences of $z$ in $y = 2 z + 7$ with $- 21$ .

$y = 2 (- 21) + 7$

$z = - 21$

$x = 15 + z$

$w = - 13 - z$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $2 (- 21) + 7$ .

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Step 8.2.1.1

Multiply $2$ by $- 21$ .

$y = - 42 + 7$

$z = - 21$

$x = 15 + z$

$w = - 13 - z$

Step 8.2.1.2

Add $- 42$ and $7$ .

$y = - 35$

$z = - 21$

$x = 15 + z$

$w = - 13 - z$

$y = - 35$

$z = - 21$

$x = 15 + z$

$w = - 13 - z$

$y = - 35$

$z = - 21$

$x = 15 + z$

$w = - 13 - z$

Step 8.3

Replace all occurrences of $z$ in $x = 15 + z$ with $- 21$ .

$x = 15 - 21$

$y = - 35$

$z = - 21$

$w = - 13 - z$

Step 8.4

Simplify $x = 15 - 21$ .

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Step 8.4.1

Simplify the left side.

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Step 8.4.1.1

Remove parentheses.

$x = 15 - 21$

$y = - 35$

$z = - 21$

$w = - 13 - z$

$x = 15 - 21$

$y = - 35$

$z = - 21$

$w = - 13 - z$

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Subtract $21$ from $15$ .

$x = - 6$

$y = - 35$

$z = - 21$

$w = - 13 - z$

$x = - 6$

$y = - 35$

$z = - 21$

$w = - 13 - z$

$x = - 6$

$y = - 35$

$z = - 21$

$w = - 13 - z$

Step 8.5

Replace all occurrences of $z$ in $w = - 13 - z$ with $- 21$ .

$w = - 13 - (- 21)$

$x = - 6$

$y = - 35$

$z = - 21$

Step 8.6

Simplify the right side.

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Step 8.6.1

Simplify $- 13 - (- 21)$ .

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Step 8.6.1.1

Multiply $- 1$ by $- 21$ .

$w = - 13 + 21$

$x = - 6$

$y = - 35$

$z = - 21$

Step 8.6.1.2

Add $- 13$ and $21$ .

$w = 8$

$x = - 6$

$y = - 35$

$z = - 21$

$w = 8$

$x = - 6$

$y = - 35$

$z = - 21$

$w = 8$

$x = - 6$

$y = - 35$

$z = - 21$

$w = 8$

$x = - 6$

$y = - 35$

$z = - 21$

Step 9

List all of the solutions.

$w = 8, x = - 6, y = - 35, z = - 21$