Precalculus Examples

$- x + y + z = - 1$ , $- x + 3 y - 5 z = - 15$ , $7 x - 6 y - 10 z = 0$

Step 1

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

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Step 1.1

Add $x$ to both sides of the equation.

$y + z = - 1 + x$

$- x + 3 y - 5 z = - 15$

$7 x - 6 y - 10 z = 0$

Step 1.2

Subtract $z$ from both sides of the equation.

$y = - 1 + x - z$

$- x + 3 y - 5 z = - 15$

$7 x - 6 y - 10 z = 0$

$y = - 1 + x - z$

$- x + 3 y - 5 z = - 15$

$7 x - 6 y - 10 z = 0$

Step 2

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

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Step 2.1

Replace all occurrences of $y$ in $- x + 3 y - 5 z = - 15$ with $- 1 + x - z$ .

$- x + 3 (- 1 + x - z) - 5 z = - 15$

$y = - 1 + x - z$

$7 x - 6 y - 10 z = 0$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- x + 3 (- 1 + x - z) - 5 z$ .

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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.

$- x + 3 \cdot - 1 + 3 x + 3 (- z) - 5 z = - 15$

$y = - 1 + x - z$

$7 x - 6 y - 10 z = 0$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $3$ by $- 1$ .

$- x - 3 + 3 x + 3 (- z) - 5 z = - 15$

$y = - 1 + x - z$

$7 x - 6 y - 10 z = 0$

Step 2.2.1.1.2.2

Multiply $- 1$ by $3$ .

$- x - 3 + 3 x - 3 z - 5 z = - 15$

$y = - 1 + x - z$

$7 x - 6 y - 10 z = 0$

$- x - 3 + 3 x - 3 z - 5 z = - 15$

$y = - 1 + x - z$

$7 x - 6 y - 10 z = 0$

$- x - 3 + 3 x - 3 z - 5 z = - 15$

$y = - 1 + x - z$

$7 x - 6 y - 10 z = 0$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $- x$ and $3 x$ .

$2 x - 3 - 3 z - 5 z = - 15$

$y = - 1 + x - z$

$7 x - 6 y - 10 z = 0$

Step 2.2.1.2.2

Subtract $5 z$ from $- 3 z$ .

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

$7 x - 6 y - 10 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

$7 x - 6 y - 10 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

$7 x - 6 y - 10 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

$7 x - 6 y - 10 z = 0$

Step 2.3

Replace all occurrences of $y$ in $7 x - 6 y - 10 z = 0$ with $- 1 + x - z$ .

$7 x - 6 (- 1 + x - z) - 10 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $7 x - 6 (- 1 + x - z) - 10 z$ .

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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.

$7 x - 6 \cdot - 1 - 6 x - 6 (- z) - 10 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 6$ by $- 1$ .

$7 x + 6 - 6 x - 6 (- z) - 10 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

Step 2.4.1.1.2.2

Multiply $- 1$ by $- 6$ .

$7 x + 6 - 6 x + 6 z - 10 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

$7 x + 6 - 6 x + 6 z - 10 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

$7 x + 6 - 6 x + 6 z - 10 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $6 x$ from $7 x$ .

$x + 6 + 6 z - 10 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

Step 2.4.1.2.2

Subtract $10 z$ from $6 z$ .

$x + 6 - 4 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

$x + 6 - 4 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

$x + 6 - 4 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

$x + 6 - 4 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

$x + 6 - 4 z = 0$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

Step 3

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

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Step 3.1

Subtract $6$ from both sides of the equation.

$x - 4 z = - 6$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

Step 3.2

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

$x = - 6 + 4 z$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

$x = - 6 + 4 z$

$2 x - 3 - 8 z = - 15$

$y = - 1 + x - z$

Step 4

Replace all occurrences of $x$ with $- 6 + 4 z$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $2 x - 3 - 8 z = - 15$ with $- 6 + 4 z$ .

$2 (- 6 + 4 z) - 3 - 8 z = - 15$

$x = - 6 + 4 z$

$y = - 1 + x - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $2 (- 6 + 4 z) - 3 - 8 z$ .

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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.

$2 \cdot - 6 + 2 (4 z) - 3 - 8 z = - 15$

$x = - 6 + 4 z$

$y = - 1 + x - z$

Step 4.2.1.1.2

Multiply $2$ by $- 6$ .

$- 12 + 2 (4 z) - 3 - 8 z = - 15$

$x = - 6 + 4 z$

$y = - 1 + x - z$

Step 4.2.1.1.3

Multiply $4$ by $2$ .

$- 12 + 8 z - 3 - 8 z = - 15$

$x = - 6 + 4 z$

$y = - 1 + x - z$

$- 12 + 8 z - 3 - 8 z = - 15$

$x = - 6 + 4 z$

$y = - 1 + x - z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $- 12 + 8 z - 3 - 8 z$ .

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Step 4.2.1.2.1.1

Subtract $8 z$ from $8 z$ .

$- 12 + 0 - 3 = - 15$

$x = - 6 + 4 z$

$y = - 1 + x - z$

Step 4.2.1.2.1.2

Add $- 12$ and $0$ .

$- 12 - 3 = - 15$

$x = - 6 + 4 z$

$y = - 1 + x - z$

$- 12 - 3 = - 15$

$x = - 6 + 4 z$

$y = - 1 + x - z$

Step 4.2.1.2.2

Subtract $3$ from $- 12$ .

$- 15 = - 15$

$x = - 6 + 4 z$

$y = - 1 + x - z$

$- 15 = - 15$

$x = - 6 + 4 z$

$y = - 1 + x - z$

$- 15 = - 15$

$x = - 6 + 4 z$

$y = - 1 + x - z$

$- 15 = - 15$

$x = - 6 + 4 z$

$y = - 1 + x - z$

Step 4.3

Replace all occurrences of $x$ in $y = - 1 + x - z$ with $- 6 + 4 z$ .

$y = - 1 - 6 + 4 z - z$

$- 15 = - 15$

$x = - 6 + 4 z$

Step 4.4

Simplify $y = - 1 - 6 + 4 z - z$ .

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Remove parentheses.

$y = - 1 - 6 + 4 z - z$

$- 15 = - 15$

$x = - 6 + 4 z$

$y = - 1 - 6 + 4 z - z$

$- 15 = - 15$

$x = - 6 + 4 z$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $- 1 - 6 + 4 z - z$ .

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Step 4.4.2.1.1

Subtract $6$ from $- 1$ .

$y = - 7 + 4 z - z$

$- 15 = - 15$

$x = - 6 + 4 z$

Step 4.4.2.1.2

Subtract $z$ from $4 z$ .

$y = - 7 + 3 z$

$- 15 = - 15$

$x = - 6 + 4 z$

$y = - 7 + 3 z$

$- 15 = - 15$

$x = - 6 + 4 z$

$y = - 7 + 3 z$

$- 15 = - 15$

$x = - 6 + 4 z$

$y = - 7 + 3 z$

$- 15 = - 15$

$x = - 6 + 4 z$

$y = - 7 + 3 z$

$- 15 = - 15$

$x = - 6 + 4 z$

Step 5

Remove any equations from the system that are always true.

$y = - 7 + 3 z$

$x = - 6 + 4 z$