Algebra Examples

$x - 3 y - z = - 3$ $- x + 8 y - 4 z = 8$ $2 x - 15 y + 7 z = - 15$

Step 1

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

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Step 1.1

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

$x - z = - 3 + 3 y$

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

$2 x - 15 y + 7 z = - 15$

Step 1.2

Add $z$ to both sides of the equation.

$x = - 3 + 3 y + z$

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

$2 x - 15 y + 7 z = - 15$

$x = - 3 + 3 y + z$

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

$2 x - 15 y + 7 z = - 15$

Step 2

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

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Step 2.1

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

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

$x = - 3 + 3 y + z$

$2 x - 15 y + 7 z = - 15$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- (- 3 + 3 y + z) + 8 y - 4 z$ .

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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.

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

$x = - 3 + 3 y + z$

$2 x - 15 y + 7 z = - 15$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $- 3$ .

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

$x = - 3 + 3 y + z$

$2 x - 15 y + 7 z = - 15$

Step 2.2.1.1.2.2

Multiply $3$ by $- 1$ .

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

$x = - 3 + 3 y + z$

$2 x - 15 y + 7 z = - 15$

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

$x = - 3 + 3 y + z$

$2 x - 15 y + 7 z = - 15$

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

$x = - 3 + 3 y + z$

$2 x - 15 y + 7 z = - 15$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $- 3 y$ and $8 y$ .

$3 + 5 y - z - 4 z = 8$

$x = - 3 + 3 y + z$

$2 x - 15 y + 7 z = - 15$

Step 2.2.1.2.2

Subtract $4 z$ from $- z$ .

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$2 x - 15 y + 7 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$2 x - 15 y + 7 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$2 x - 15 y + 7 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$2 x - 15 y + 7 z = - 15$

Step 2.3

Replace all occurrences of $x$ in $2 x - 15 y + 7 z = - 15$ with $- 3 + 3 y + z$ .

$2 (- 3 + 3 y + z) - 15 y + 7 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $2 (- 3 + 3 y + z) - 15 y + 7 z$ .

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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.

$2 \cdot - 3 + 2 (3 y) + 2 z - 15 y + 7 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $2$ by $- 3$ .

$- 6 + 2 (3 y) + 2 z - 15 y + 7 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

Step 2.4.1.1.2.2

Multiply $3$ by $2$ .

$- 6 + 6 y + 2 z - 15 y + 7 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$- 6 + 6 y + 2 z - 15 y + 7 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$- 6 + 6 y + 2 z - 15 y + 7 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $15 y$ from $6 y$ .

$- 6 - 9 y + 2 z + 7 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

Step 2.4.1.2.2

Add $2 z$ and $7 z$ .

$- 6 - 9 y + 9 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$- 6 - 9 y + 9 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$- 6 - 9 y + 9 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$- 6 - 9 y + 9 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$- 6 - 9 y + 9 z = - 15$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

Step 3

Solve for $y$ in $- 6 - 9 y + 9 z = - 15$ .

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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 $6$ to both sides of the equation.

$- 9 y + 9 z = - 15 + 6$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

Step 3.1.2

Subtract $9 z$ from both sides of the equation.

$- 9 y = - 15 + 6 - 9 z$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

Step 3.1.3

Add $- 15$ and $6$ .

$- 9 y = - 9 - 9 z$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$- 9 y = - 9 - 9 z$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

Step 3.2

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

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Step 3.2.1

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

$\frac{- 9 y}{- 9} = \frac{- 9}{- 9} + \frac{- 9 z}{- 9}$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

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.

$\frac{- 9 y}{- 9} = \frac{- 9}{- 9} + \frac{- 9 z}{- 9}$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 9}{- 9} + \frac{- 9 z}{- 9}$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$y = \frac{- 9}{- 9} + \frac{- 9 z}{- 9}$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$y = \frac{- 9}{- 9} + \frac{- 9 z}{- 9}$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

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 $- 9$ by $- 9$ .

$y = 1 + \frac{- 9 z}{- 9}$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

Step 3.2.3.1.2

Cancel the common factor of $- 9$ .

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Step 3.2.3.1.2.1

Cancel the common factor.

$y = 1 + \frac{- 9 z}{- 9}$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

Step 3.2.3.1.2.2

Divide $z$ by $1$ .

$y = 1 + z$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$y = 1 + z$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$y = 1 + z$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$y = 1 + z$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$y = 1 + z$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

$y = 1 + z$

$3 + 5 y - 5 z = 8$

$x = - 3 + 3 y + z$

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $3 + 5 y - 5 z = 8$ with $1 + z$ .

$3 + 5 (1 + z) - 5 z = 8$

$y = 1 + z$

$x = - 3 + 3 y + z$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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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.

$3 + 5 \cdot 1 + 5 z - 5 z = 8$

$y = 1 + z$

$x = - 3 + 3 y + z$

Step 4.2.1.1.2

Multiply $5$ by $1$ .

$3 + 5 + 5 z - 5 z = 8$

$y = 1 + z$

$x = - 3 + 3 y + z$

$3 + 5 + 5 z - 5 z = 8$

$y = 1 + z$

$x = - 3 + 3 y + z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $3 + 5 + 5 z - 5 z$ .

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Step 4.2.1.2.1.1

Subtract $5 z$ from $5 z$ .

$3 + 5 + 0 = 8$

$y = 1 + z$

$x = - 3 + 3 y + z$

Step 4.2.1.2.1.2

Add $3 + 5$ and $0$ .

$3 + 5 = 8$

$y = 1 + z$

$x = - 3 + 3 y + z$

$3 + 5 = 8$

$y = 1 + z$

$x = - 3 + 3 y + z$

Step 4.2.1.2.2

Add $3$ and $5$ .

$8 = 8$

$y = 1 + z$

$x = - 3 + 3 y + z$

$8 = 8$

$y = 1 + z$

$x = - 3 + 3 y + z$

$8 = 8$

$y = 1 + z$

$x = - 3 + 3 y + z$

$8 = 8$

$y = 1 + z$

$x = - 3 + 3 y + z$

Step 4.3

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

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

$8 = 8$

$y = 1 + z$

Step 4.4

Simplify the right side.

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Step 4.4.1

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

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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.

$x = - 3 + 3 \cdot 1 + 3 z + z$

$8 = 8$

$y = 1 + z$

Step 4.4.1.1.2

Multiply $3$ by $1$ .

$x = - 3 + 3 + 3 z + z$

$8 = 8$

$y = 1 + z$

$x = - 3 + 3 + 3 z + z$

$8 = 8$

$y = 1 + z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Add $- 3$ and $3$ .

$x = 0 + 3 z + z$

$8 = 8$

$y = 1 + z$

Step 4.4.1.2.2

Add $0$ and $3 z$ .

$x = 3 z + z$

$8 = 8$

$y = 1 + z$

Step 4.4.1.2.3

Add $3 z$ and $z$ .

$x = 4 z$

$8 = 8$

$y = 1 + z$

$x = 4 z$

$8 = 8$

$y = 1 + z$

$x = 4 z$

$8 = 8$

$y = 1 + z$

$x = 4 z$

$8 = 8$

$y = 1 + z$

$x = 4 z$

$8 = 8$

$y = 1 + z$

Step 5

Remove any equations from the system that are always true.

$x = 4 z$

$y = 1 + z$