Algebra Examples

$- 8 x + 7 y - 8 z = - 9$ $- 8 x + 7 y - 10 z = - 7$ $x - 2 y + z = 9$

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.

$x + z = 9 + 2 y$

$- 8 x + 7 y - 8 z = - 9$

$- 8 x + 7 y - 10 z = - 7$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 9 + 2 y - z$

$- 8 x + 7 y - 8 z = - 9$

$- 8 x + 7 y - 10 z = - 7$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 8 z = - 9$

$- 8 x + 7 y - 10 z = - 7$

Step 2

Replace all occurrences of $x$ with $9 + 2 y - z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- 8 x + 7 y - 8 z = - 9$ with $9 + 2 y - z$ .

$- 8 (9 + 2 y - z) + 7 y - 8 z = - 9$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 10 z = - 7$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 8 (9 + 2 y - z) + 7 y - 8 z$ .

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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.

$- 8 \cdot 9 - 8 (2 y) - 8 (- z) + 7 y - 8 z = - 9$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 10 z = - 7$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 8$ by $9$ .

$- 72 - 8 (2 y) - 8 (- z) + 7 y - 8 z = - 9$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 10 z = - 7$

Step 2.2.1.1.2.2

Multiply $2$ by $- 8$ .

$- 72 - 16 y - 8 (- z) + 7 y - 8 z = - 9$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 10 z = - 7$

Step 2.2.1.1.2.3

Multiply $- 1$ by $- 8$ .

$- 72 - 16 y + 8 z + 7 y - 8 z = - 9$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 10 z = - 7$

$- 72 - 16 y + 8 z + 7 y - 8 z = - 9$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 10 z = - 7$

$- 72 - 16 y + 8 z + 7 y - 8 z = - 9$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 10 z = - 7$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Combine the opposite terms in $- 72 - 16 y + 8 z + 7 y - 8 z$ .

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Step 2.2.1.2.1.1

Subtract $8 z$ from $8 z$ .

$- 72 - 16 y + 7 y + 0 = - 9$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 10 z = - 7$

Step 2.2.1.2.1.2

Add $- 72 - 16 y + 7 y$ and $0$ .

$- 72 - 16 y + 7 y = - 9$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 10 z = - 7$

$- 72 - 16 y + 7 y = - 9$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 10 z = - 7$

Step 2.2.1.2.2

Add $- 16 y$ and $7 y$ .

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 10 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 10 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 10 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

$- 8 x + 7 y - 10 z = - 7$

Step 2.3

Replace all occurrences of $x$ in $- 8 x + 7 y - 10 z = - 7$ with $9 + 2 y - z$ .

$- 8 (9 + 2 y - z) + 7 y - 10 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- 8 (9 + 2 y - z) + 7 y - 10 z$ .

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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.

$- 8 \cdot 9 - 8 (2 y) - 8 (- z) + 7 y - 10 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 8$ by $9$ .

$- 72 - 8 (2 y) - 8 (- z) + 7 y - 10 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

Step 2.4.1.1.2.2

Multiply $2$ by $- 8$ .

$- 72 - 16 y - 8 (- z) + 7 y - 10 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

Step 2.4.1.1.2.3

Multiply $- 1$ by $- 8$ .

$- 72 - 16 y + 8 z + 7 y - 10 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

$- 72 - 16 y + 8 z + 7 y - 10 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

$- 72 - 16 y + 8 z + 7 y - 10 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $- 16 y$ and $7 y$ .

$- 72 - 9 y + 8 z - 10 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

Step 2.4.1.2.2

Subtract $10 z$ from $8 z$ .

$- 72 - 9 y - 2 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

$- 72 - 9 y - 2 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

$- 72 - 9 y - 2 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

$- 72 - 9 y - 2 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

$- 72 - 9 y - 2 z = - 7$

$- 72 - 9 y = - 9$

$x = 9 + 2 y - z$

Step 3

Solve for $y$ in $- 72 - 9 y = - 9$ .

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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

Add $72$ to both sides of the equation.

$- 9 y = - 9 + 72$

$- 72 - 9 y - 2 z = - 7$

$x = 9 + 2 y - z$

Step 3.1.2

Add $- 9$ and $72$ .

$- 9 y = 63$

$- 72 - 9 y - 2 z = - 7$

$x = 9 + 2 y - z$

$- 9 y = 63$

$- 72 - 9 y - 2 z = - 7$

$x = 9 + 2 y - z$

Step 3.2

Divide each term in $- 9 y = 63$ by $- 9$ and simplify.

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Step 3.2.1

Divide each term in $- 9 y = 63$ by $- 9$ .

$\frac{- 9 y}{- 9} = \frac{63}{- 9}$

$- 72 - 9 y - 2 z = - 7$

$x = 9 + 2 y - z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 9$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 9 y}{- 9} = \frac{63}{- 9}$

$- 72 - 9 y - 2 z = - 7$

$x = 9 + 2 y - z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{63}{- 9}$

$- 72 - 9 y - 2 z = - 7$

$x = 9 + 2 y - z$

$y = \frac{63}{- 9}$

$- 72 - 9 y - 2 z = - 7$

$x = 9 + 2 y - z$

$y = \frac{63}{- 9}$

$- 72 - 9 y - 2 z = - 7$

$x = 9 + 2 y - z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $63$ by $- 9$ .

$y = - 7$

$- 72 - 9 y - 2 z = - 7$

$x = 9 + 2 y - z$

$y = - 7$

$- 72 - 9 y - 2 z = - 7$

$x = 9 + 2 y - z$

$y = - 7$

$- 72 - 9 y - 2 z = - 7$

$x = 9 + 2 y - z$

$y = - 7$

$- 72 - 9 y - 2 z = - 7$

$x = 9 + 2 y - z$

Step 4

Replace all occurrences of $y$ with $- 7$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $- 72 - 9 y - 2 z = - 7$ with $- 7$ .

$- 72 - 9 \cdot - 7 - 2 z = - 7$

$y = - 7$

$x = 9 + 2 y - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 72 - 9 \cdot - 7 - 2 z$ .

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Step 4.2.1.1

Multiply $- 9$ by $- 7$ .

$- 72 + 63 - 2 z = - 7$

$y = - 7$

$x = 9 + 2 y - z$

Step 4.2.1.2

Add $- 72$ and $63$ .

$- 9 - 2 z = - 7$

$y = - 7$

$x = 9 + 2 y - z$

$- 9 - 2 z = - 7$

$y = - 7$

$x = 9 + 2 y - z$

$- 9 - 2 z = - 7$

$y = - 7$

$x = 9 + 2 y - z$

Step 4.3

Replace all occurrences of $y$ in $x = 9 + 2 y - z$ with $- 7$ .

$x = 9 + 2 (- 7) - z$

$- 9 - 2 z = - 7$

$y = - 7$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $9 + 2 (- 7) - z$ .

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Step 4.4.1.1

Multiply $2$ by $- 7$ .

$x = 9 - 14 - z$

$- 9 - 2 z = - 7$

$y = - 7$

Step 4.4.1.2

Subtract $14$ from $9$ .

$x = - 5 - z$

$- 9 - 2 z = - 7$

$y = - 7$

$x = - 5 - z$

$- 9 - 2 z = - 7$

$y = - 7$

$x = - 5 - z$

$- 9 - 2 z = - 7$

$y = - 7$

$x = - 5 - z$

$- 9 - 2 z = - 7$

$y = - 7$

Step 5

Solve for $z$ in $- 9 - 2 z = - 7$ .

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Step 5.1

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

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Step 5.1.1

Add $9$ to both sides of the equation.

$- 2 z = - 7 + 9$

$x = - 5 - z$

$y = - 7$

Step 5.1.2

Add $- 7$ and $9$ .

$- 2 z = 2$

$x = - 5 - z$

$y = - 7$

$- 2 z = 2$

$x = - 5 - z$

$y = - 7$

Step 5.2

Divide each term in $- 2 z = 2$ by $- 2$ and simplify.

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Step 5.2.1

Divide each term in $- 2 z = 2$ by $- 2$ .

$\frac{- 2 z}{- 2} = \frac{2}{- 2}$

$x = - 5 - z$

$y = - 7$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 2 z}{- 2} = \frac{2}{- 2}$

$x = - 5 - z$

$y = - 7$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{2}{- 2}$

$x = - 5 - z$

$y = - 7$

$z = \frac{2}{- 2}$

$x = - 5 - z$

$y = - 7$

$z = \frac{2}{- 2}$

$x = - 5 - z$

$y = - 7$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $2$ by $- 2$ .

$z = - 1$

$x = - 5 - z$

$y = - 7$

$z = - 1$

$x = - 5 - z$

$y = - 7$

$z = - 1$

$x = - 5 - z$

$y = - 7$

$z = - 1$

$x = - 5 - z$

$y = - 7$

Step 6

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

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Step 6.1

Replace all occurrences of $z$ in $x = - 5 - z$ with $- 1$ .

$x = - 5 - (- 1)$

$z = - 1$

$y = - 7$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 5 - (- 1)$ .

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Step 6.2.1.1

Multiply $- 1$ by $- 1$ .

$x = - 5 + 1$

$z = - 1$

$y = - 7$

Step 6.2.1.2

Add $- 5$ and $1$ .

$x = - 4$

$z = - 1$

$y = - 7$

$x = - 4$

$z = - 1$

$y = - 7$

$x = - 4$

$z = - 1$

$y = - 7$

$x = - 4$

$z = - 1$

$y = - 7$

Step 7

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- 4, - 7, - 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 4, - 7, - 1)$

Equation Form:

$x = - 4, y = - 7, z = - 1$