Algebra Examples

$- 2 x - 3 y + z = - 6$ $x + y - z = 5$ $7 x + 8 y - 6 z = 31$

Step 1

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

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Step 1.1

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

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

$x + y - z = 5$

$7 x + 8 y - 6 z = 31$

Step 1.2

Add $3 y$ to both sides of the equation.

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

$x + y - z = 5$

$7 x + 8 y - 6 z = 31$

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

$x + y - z = 5$

$7 x + 8 y - 6 z = 31$

Step 2

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

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Step 2.1

Replace all occurrences of $z$ in $x + y - z = 5$ with $- 6 + 2 x + 3 y$ .

$x + y - (- 6 + 2 x + 3 y) = 5$

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

$7 x + 8 y - 6 z = 31$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$x + y + 6 - (2 x) - (3 y) = 5$

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

$7 x + 8 y - 6 z = 31$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $- 6$ .

$x + y + 6 - (2 x) - (3 y) = 5$

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

$7 x + 8 y - 6 z = 31$

Step 2.2.1.1.2.2

Multiply $2$ by $- 1$ .

$x + y + 6 - 2 x - (3 y) = 5$

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

$7 x + 8 y - 6 z = 31$

Step 2.2.1.1.2.3

Multiply $3$ by $- 1$ .

$x + y + 6 - 2 x - 3 y = 5$

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

$7 x + 8 y - 6 z = 31$

$x + y + 6 - 2 x - 3 y = 5$

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

$7 x + 8 y - 6 z = 31$

$x + y + 6 - 2 x - 3 y = 5$

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

$7 x + 8 y - 6 z = 31$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $2 x$ from $x$ .

$- x + y + 6 - 3 y = 5$

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

$7 x + 8 y - 6 z = 31$

Step 2.2.1.2.2

Subtract $3 y$ from $y$ .

$- x - 2 y + 6 = 5$

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

$7 x + 8 y - 6 z = 31$

$- x - 2 y + 6 = 5$

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

$7 x + 8 y - 6 z = 31$

$- x - 2 y + 6 = 5$

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

$7 x + 8 y - 6 z = 31$

$- x - 2 y + 6 = 5$

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

$7 x + 8 y - 6 z = 31$

Step 2.3

Replace all occurrences of $z$ in $7 x + 8 y - 6 z = 31$ with $- 6 + 2 x + 3 y$ .

$7 x + 8 y - 6 (- 6 + 2 x + 3 y) = 31$

$- x - 2 y + 6 = 5$

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

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $7 x + 8 y - 6 (- 6 + 2 x + 3 y)$ .

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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.

$7 x + 8 y - 6 \cdot - 6 - 6 (2 x) - 6 (3 y) = 31$

$- x - 2 y + 6 = 5$

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

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 6$ by $- 6$ .

$7 x + 8 y + 36 - 6 (2 x) - 6 (3 y) = 31$

$- x - 2 y + 6 = 5$

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

Step 2.4.1.1.2.2

Multiply $2$ by $- 6$ .

$7 x + 8 y + 36 - 12 x - 6 (3 y) = 31$

$- x - 2 y + 6 = 5$

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

Step 2.4.1.1.2.3

Multiply $3$ by $- 6$ .

$7 x + 8 y + 36 - 12 x - 18 y = 31$

$- x - 2 y + 6 = 5$

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

$7 x + 8 y + 36 - 12 x - 18 y = 31$

$- x - 2 y + 6 = 5$

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

$7 x + 8 y + 36 - 12 x - 18 y = 31$

$- x - 2 y + 6 = 5$

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

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $12 x$ from $7 x$ .

$- 5 x + 8 y + 36 - 18 y = 31$

$- x - 2 y + 6 = 5$

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

Step 2.4.1.2.2

Subtract $18 y$ from $8 y$ .

$- 5 x - 10 y + 36 = 31$

$- x - 2 y + 6 = 5$

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

$- 5 x - 10 y + 36 = 31$

$- x - 2 y + 6 = 5$

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

$- 5 x - 10 y + 36 = 31$

$- x - 2 y + 6 = 5$

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

$- 5 x - 10 y + 36 = 31$

$- x - 2 y + 6 = 5$

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

$- 5 x - 10 y + 36 = 31$

$- x - 2 y + 6 = 5$

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

Step 3

Solve for $x$ in $- 5 x - 10 y + 36 = 31$ .

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Step 3.1

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

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Step 3.1.1

Add $10 y$ to both sides of the equation.

$- 5 x + 36 = 31 + 10 y$

$- x - 2 y + 6 = 5$

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

Step 3.1.2

Subtract $36$ from both sides of the equation.

$- 5 x = 31 + 10 y - 36$

$- x - 2 y + 6 = 5$

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

Step 3.1.3

Subtract $36$ from $31$ .

$- 5 x = 10 y - 5$

$- x - 2 y + 6 = 5$

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

$- 5 x = 10 y - 5$

$- x - 2 y + 6 = 5$

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

Step 3.2

Divide each term in $- 5 x = 10 y - 5$ by $- 5$ and simplify.

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Step 3.2.1

Divide each term in $- 5 x = 10 y - 5$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{10 y}{- 5} + \frac{- 5}{- 5}$

$- x - 2 y + 6 = 5$

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{10 y}{- 5} + \frac{- 5}{- 5}$

$- x - 2 y + 6 = 5$

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

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10 y}{- 5} + \frac{- 5}{- 5}$

$- x - 2 y + 6 = 5$

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

$x = \frac{10 y}{- 5} + \frac{- 5}{- 5}$

$- x - 2 y + 6 = 5$

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

$x = \frac{10 y}{- 5} + \frac{- 5}{- 5}$

$- x - 2 y + 6 = 5$

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

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

Cancel the common factor of $10$ and $- 5$ .

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Step 3.2.3.1.1.1

Factor $5$ out of $10 y$ .

$x = \frac{5 (2 y)}{- 5} + \frac{- 5}{- 5}$

$- x - 2 y + 6 = 5$

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

Step 3.2.3.1.1.2

Move the negative one from the denominator of $\frac{2 y}{- 1}$ .

$x = - 1 \cdot (2 y) + \frac{- 5}{- 5}$

$- x - 2 y + 6 = 5$

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

$x = - 1 \cdot (2 y) + \frac{- 5}{- 5}$

$- x - 2 y + 6 = 5$

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

Step 3.2.3.1.2

Rewrite $- 1 \cdot (2 y)$ as $- (2 y)$ .

$x = - (2 y) + \frac{- 5}{- 5}$

$- x - 2 y + 6 = 5$

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

Step 3.2.3.1.3

Multiply $2$ by $- 1$ .

$x = - 2 y + \frac{- 5}{- 5}$

$- x - 2 y + 6 = 5$

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

Step 3.2.3.1.4

Divide $- 5$ by $- 5$ .

$x = - 2 y + 1$

$- x - 2 y + 6 = 5$

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

$x = - 2 y + 1$

$- x - 2 y + 6 = 5$

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

$x = - 2 y + 1$

$- x - 2 y + 6 = 5$

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

$x = - 2 y + 1$

$- x - 2 y + 6 = 5$

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

$x = - 2 y + 1$

$- x - 2 y + 6 = 5$

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

Step 4

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

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Step 4.1

Replace all occurrences of $x$ in $- x - 2 y + 6 = 5$ with $- 2 y + 1$ .

$- (- 2 y + 1) - 2 y + 6 = 5$

$x = - 2 y + 1$

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- (- 2 y + 1) - 2 y + 6$ .

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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.

$- (- 2 y) - 1 \cdot 1 - 2 y + 6 = 5$

$x = - 2 y + 1$

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

Step 4.2.1.1.2

Multiply $- 2$ by $- 1$ .

$2 y - 1 \cdot 1 - 2 y + 6 = 5$

$x = - 2 y + 1$

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

Step 4.2.1.1.3

Multiply $- 1$ by $1$ .

$2 y - 1 - 2 y + 6 = 5$

$x = - 2 y + 1$

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

$2 y - 1 - 2 y + 6 = 5$

$x = - 2 y + 1$

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

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $2 y - 1 - 2 y + 6$ .

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Step 4.2.1.2.1.1

Subtract $2 y$ from $2 y$ .

$0 - 1 + 6 = 5$

$x = - 2 y + 1$

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

Step 4.2.1.2.1.2

Subtract $1$ from $0$ .

$- 1 + 6 = 5$

$x = - 2 y + 1$

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

$- 1 + 6 = 5$

$x = - 2 y + 1$

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

Step 4.2.1.2.2

Add $- 1$ and $6$ .

$5 = 5$

$x = - 2 y + 1$

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

$5 = 5$

$x = - 2 y + 1$

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

$5 = 5$

$x = - 2 y + 1$

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

$5 = 5$

$x = - 2 y + 1$

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

Step 4.3

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

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

$5 = 5$

$x = - 2 y + 1$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- 6 + 2 (- 2 y + 1) + 3 y$ .

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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.

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

$5 = 5$

$x = - 2 y + 1$

Step 4.4.1.1.2

Multiply $- 2$ by $2$ .

$z = - 6 - 4 y + 2 \cdot 1 + 3 y$

$5 = 5$

$x = - 2 y + 1$

Step 4.4.1.1.3

Multiply $2$ by $1$ .

$z = - 6 - 4 y + 2 + 3 y$

$5 = 5$

$x = - 2 y + 1$

$z = - 6 - 4 y + 2 + 3 y$

$5 = 5$

$x = - 2 y + 1$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Add $- 6$ and $2$ .

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

$5 = 5$

$x = - 2 y + 1$

Step 4.4.1.2.2

Add $- 4 y$ and $3 y$ .

$z = - y - 4$

$5 = 5$

$x = - 2 y + 1$

$z = - y - 4$

$5 = 5$

$x = - 2 y + 1$

$z = - y - 4$

$5 = 5$

$x = - 2 y + 1$

$z = - y - 4$

$5 = 5$

$x = - 2 y + 1$

$z = - y - 4$

$5 = 5$

$x = - 2 y + 1$

Step 5

Remove any equations from the system that are always true.

$z = - y - 4$

$x = - 2 y + 1$