Finite Math Examples

$x - 2 y - z = 6$ , $- x + 6 y - 3 z = - 18$ , $2 x - 11 y + 5 z = 33$

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.

$x - z = 6 + 2 y$

$- x + 6 y - 3 z = - 18$

$2 x - 11 y + 5 z = 33$

Step 1.2

Add $z$ to both sides of the equation.

$x = 6 + 2 y + z$

$- x + 6 y - 3 z = - 18$

$2 x - 11 y + 5 z = 33$

$x = 6 + 2 y + z$

$- x + 6 y - 3 z = - 18$

$2 x - 11 y + 5 z = 33$

Step 2

Replace all occurrences of $x$ with $6 + 2 y + z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- x + 6 y - 3 z = - 18$ with $6 + 2 y + z$ .

$- (6 + 2 y + z) + 6 y - 3 z = - 18$

$x = 6 + 2 y + z$

$2 x - 11 y + 5 z = 33$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$- 1 \cdot 6 - (2 y) - z + 6 y - 3 z = - 18$

$x = 6 + 2 y + z$

$2 x - 11 y + 5 z = 33$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $6$ .

$- 6 - (2 y) - z + 6 y - 3 z = - 18$

$x = 6 + 2 y + z$

$2 x - 11 y + 5 z = 33$

Step 2.2.1.1.2.2

Multiply $2$ by $- 1$ .

$- 6 - 2 y - z + 6 y - 3 z = - 18$

$x = 6 + 2 y + z$

$2 x - 11 y + 5 z = 33$

$- 6 - 2 y - z + 6 y - 3 z = - 18$

$x = 6 + 2 y + z$

$2 x - 11 y + 5 z = 33$

$- 6 - 2 y - z + 6 y - 3 z = - 18$

$x = 6 + 2 y + z$

$2 x - 11 y + 5 z = 33$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $- 2 y$ and $6 y$ .

$- 6 + 4 y - z - 3 z = - 18$

$x = 6 + 2 y + z$

$2 x - 11 y + 5 z = 33$

Step 2.2.1.2.2

Subtract $3 z$ from $- z$ .

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$2 x - 11 y + 5 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$2 x - 11 y + 5 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$2 x - 11 y + 5 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$2 x - 11 y + 5 z = 33$

Step 2.3

Replace all occurrences of $x$ in $2 x - 11 y + 5 z = 33$ with $6 + 2 y + z$ .

$2 (6 + 2 y + z) - 11 y + 5 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $2 (6 + 2 y + z) - 11 y + 5 z$ .

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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.

$2 \cdot 6 + 2 (2 y) + 2 z - 11 y + 5 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $2$ by $6$ .

$12 + 2 (2 y) + 2 z - 11 y + 5 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 2.4.1.1.2.2

Multiply $2$ by $2$ .

$12 + 4 y + 2 z - 11 y + 5 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$12 + 4 y + 2 z - 11 y + 5 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$12 + 4 y + 2 z - 11 y + 5 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $11 y$ from $4 y$ .

$12 - 7 y + 2 z + 5 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 2.4.1.2.2

Add $2 z$ and $5 z$ .

$12 - 7 y + 7 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$12 - 7 y + 7 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$12 - 7 y + 7 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$12 - 7 y + 7 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$12 - 7 y + 7 z = 33$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 3

Solve for $y$ in $12 - 7 y + 7 z = 33$ .

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Step 3.1

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

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Step 3.1.1

Subtract $12$ from both sides of the equation.

$- 7 y + 7 z = 33 - 12$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 3.1.2

Subtract $7 z$ from both sides of the equation.

$- 7 y = 33 - 12 - 7 z$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 3.1.3

Subtract $12$ from $33$ .

$- 7 y = 21 - 7 z$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$- 7 y = 21 - 7 z$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 3.2

Divide each term in $- 7 y = 21 - 7 z$ by $- 7$ and simplify.

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Step 3.2.1

Divide each term in $- 7 y = 21 - 7 z$ by $- 7$ .

$\frac{- 7 y}{- 7} = \frac{21}{- 7} + \frac{- 7 z}{- 7}$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 7$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 7 y}{- 7} = \frac{21}{- 7} + \frac{- 7 z}{- 7}$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{21}{- 7} + \frac{- 7 z}{- 7}$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$y = \frac{21}{- 7} + \frac{- 7 z}{- 7}$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$y = \frac{21}{- 7} + \frac{- 7 z}{- 7}$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Divide $21$ by $- 7$ .

$y = - 3 + \frac{- 7 z}{- 7}$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 3.2.3.1.2

Cancel the common factor of $- 7$ .

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Step 3.2.3.1.2.1

Cancel the common factor.

$y = - 3 + \frac{- 7 z}{- 7}$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 3.2.3.1.2.2

Divide $z$ by $1$ .

$y = - 3 + z$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$y = - 3 + z$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$y = - 3 + z$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$y = - 3 + z$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$y = - 3 + z$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

$y = - 3 + z$

$- 6 + 4 y - 4 z = - 18$

$x = 6 + 2 y + z$

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $- 6 + 4 y - 4 z = - 18$ with $- 3 + z$ .

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

$y = - 3 + z$

$x = 6 + 2 y + z$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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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.

$- 6 + 4 \cdot - 3 + 4 z - 4 z = - 18$

$y = - 3 + z$

$x = 6 + 2 y + z$

Step 4.2.1.1.2

Multiply $4$ by $- 3$ .

$- 6 - 12 + 4 z - 4 z = - 18$

$y = - 3 + z$

$x = 6 + 2 y + z$

$- 6 - 12 + 4 z - 4 z = - 18$

$y = - 3 + z$

$x = 6 + 2 y + z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $- 6 - 12 + 4 z - 4 z$ .

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Step 4.2.1.2.1.1

Subtract $4 z$ from $4 z$ .

$- 6 - 12 + 0 = - 18$

$y = - 3 + z$

$x = 6 + 2 y + z$

Step 4.2.1.2.1.2

Add $- 6 - 12$ and $0$ .

$- 6 - 12 = - 18$

$y = - 3 + z$

$x = 6 + 2 y + z$

$- 6 - 12 = - 18$

$y = - 3 + z$

$x = 6 + 2 y + z$

Step 4.2.1.2.2

Subtract $12$ from $- 6$ .

$- 18 = - 18$

$y = - 3 + z$

$x = 6 + 2 y + z$

$- 18 = - 18$

$y = - 3 + z$

$x = 6 + 2 y + z$

$- 18 = - 18$

$y = - 3 + z$

$x = 6 + 2 y + z$

$- 18 = - 18$

$y = - 3 + z$

$x = 6 + 2 y + z$

Step 4.3

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

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

$- 18 = - 18$

$y = - 3 + z$

Step 4.4

Simplify the right side.

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Step 4.4.1

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

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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.

$x = 6 + 2 \cdot - 3 + 2 z + z$

$- 18 = - 18$

$y = - 3 + z$

Step 4.4.1.1.2

Multiply $2$ by $- 3$ .

$x = 6 - 6 + 2 z + z$

$- 18 = - 18$

$y = - 3 + z$

$x = 6 - 6 + 2 z + z$

$- 18 = - 18$

$y = - 3 + z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $6$ from $6$ .

$x = 0 + 2 z + z$

$- 18 = - 18$

$y = - 3 + z$

Step 4.4.1.2.2

Add $0$ and $2 z$ .

$x = 2 z + z$

$- 18 = - 18$

$y = - 3 + z$

Step 4.4.1.2.3

Add $2 z$ and $z$ .

$x = 3 z$

$- 18 = - 18$

$y = - 3 + z$

$x = 3 z$

$- 18 = - 18$

$y = - 3 + z$

$x = 3 z$

$- 18 = - 18$

$y = - 3 + z$

$x = 3 z$

$- 18 = - 18$

$y = - 3 + z$

$x = 3 z$

$- 18 = - 18$

$y = - 3 + z$

Step 5

Remove any equations from the system that are always true.

$x = 3 z$

$y = - 3 + z$