Algebra Examples

$3 x - 9 y - 3 z = 6$ $2 x - 7 y - 4 z = 8$ $4 x - 8 y + 4 z = - 8$

Step 1

Solve for $x$ in $3 x - 9 y - 3 z = 6$ .

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Step 1.1

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

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Step 1.1.1

Add $9 y$ to both sides of the equation.

$3 x - 3 z = 6 + 9 y$

$2 x - 7 y - 4 z = 8$

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

Step 1.1.2

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

$3 x = 6 + 9 y + 3 z$

$2 x - 7 y - 4 z = 8$

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

$3 x = 6 + 9 y + 3 z$

$2 x - 7 y - 4 z = 8$

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

Step 1.2

Divide each term in $3 x = 6 + 9 y + 3 z$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 6 + 9 y + 3 z$ by $3$ .

$\frac{3 x}{3} = \frac{6}{3} + \frac{9 y}{3} + \frac{3 z}{3}$

$2 x - 7 y - 4 z = 8$

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{6}{3} + \frac{9 y}{3} + \frac{3 z}{3}$

$2 x - 7 y - 4 z = 8$

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

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{3} + \frac{9 y}{3} + \frac{3 z}{3}$

$2 x - 7 y - 4 z = 8$

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

$x = \frac{6}{3} + \frac{9 y}{3} + \frac{3 z}{3}$

$2 x - 7 y - 4 z = 8$

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

$x = \frac{6}{3} + \frac{9 y}{3} + \frac{3 z}{3}$

$2 x - 7 y - 4 z = 8$

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $6$ by $3$ .

$x = 2 + \frac{9 y}{3} + \frac{3 z}{3}$

$2 x - 7 y - 4 z = 8$

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

Step 1.2.3.1.2

Cancel the common factor of $9$ and $3$ .

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Step 1.2.3.1.2.1

Factor $3$ out of $9 y$ .

$x = 2 + \frac{3 (3 y)}{3} + \frac{3 z}{3}$

$2 x - 7 y - 4 z = 8$

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

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $3$ out of $3$ .

$x = 2 + \frac{3 (3 y)}{3 (1)} + \frac{3 z}{3}$

$2 x - 7 y - 4 z = 8$

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

Step 1.2.3.1.2.2.2

Cancel the common factor.

$x = 2 + \frac{3 (3 y)}{3 \cdot 1} + \frac{3 z}{3}$

$2 x - 7 y - 4 z = 8$

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

Step 1.2.3.1.2.2.3

Rewrite the expression.

$x = 2 + \frac{3 y}{1} + \frac{3 z}{3}$

$2 x - 7 y - 4 z = 8$

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

Step 1.2.3.1.2.2.4

Divide $3 y$ by $1$ .

$x = 2 + 3 y + \frac{3 z}{3}$

$2 x - 7 y - 4 z = 8$

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

$x = 2 + 3 y + \frac{3 z}{3}$

$2 x - 7 y - 4 z = 8$

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

$x = 2 + 3 y + \frac{3 z}{3}$

$2 x - 7 y - 4 z = 8$

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

Step 1.2.3.1.3

Cancel the common factor of $3$ .

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Step 1.2.3.1.3.1

Cancel the common factor.

$x = 2 + 3 y + \frac{3 z}{3}$

$2 x - 7 y - 4 z = 8$

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

Step 1.2.3.1.3.2

Divide $z$ by $1$ .

$x = 2 + 3 y + z$

$2 x - 7 y - 4 z = 8$

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

$x = 2 + 3 y + z$

$2 x - 7 y - 4 z = 8$

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

$x = 2 + 3 y + z$

$2 x - 7 y - 4 z = 8$

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

$x = 2 + 3 y + z$

$2 x - 7 y - 4 z = 8$

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

$x = 2 + 3 y + z$

$2 x - 7 y - 4 z = 8$

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

$x = 2 + 3 y + z$

$2 x - 7 y - 4 z = 8$

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

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $2 x - 7 y - 4 z = 8$ with $2 + 3 y + z$ .

$2 (2 + 3 y + z) - 7 y - 4 z = 8$

$x = 2 + 3 y + z$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (2 + 3 y + z) - 7 y - 4 z$ .

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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.

$2 \cdot 2 + 2 (3 y) + 2 z - 7 y - 4 z = 8$

$x = 2 + 3 y + z$

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

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $2$ by $2$ .

$4 + 2 (3 y) + 2 z - 7 y - 4 z = 8$

$x = 2 + 3 y + z$

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

Step 2.2.1.1.2.2

Multiply $3$ by $2$ .

$4 + 6 y + 2 z - 7 y - 4 z = 8$

$x = 2 + 3 y + z$

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

$4 + 6 y + 2 z - 7 y - 4 z = 8$

$x = 2 + 3 y + z$

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

$4 + 6 y + 2 z - 7 y - 4 z = 8$

$x = 2 + 3 y + z$

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

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $7 y$ from $6 y$ .

$4 - y + 2 z - 4 z = 8$

$x = 2 + 3 y + z$

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

Step 2.2.1.2.2

Subtract $4 z$ from $2 z$ .

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

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

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

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

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

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

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

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

Step 2.3

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

$4 (2 + 3 y + z) - 8 y + 4 z = - 8$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

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

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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.

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

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $4$ by $2$ .

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

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 2.4.1.1.2.2

Multiply $3$ by $4$ .

$8 + 12 y + 4 z - 8 y + 4 z = - 8$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

$8 + 12 y + 4 z - 8 y + 4 z = - 8$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

$8 + 12 y + 4 z - 8 y + 4 z = - 8$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $8 y$ from $12 y$ .

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

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 2.4.1.2.2

Add $4 z$ and $4 z$ .

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

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

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

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

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

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

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

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

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

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 3

Solve for $y$ in $8 + 4 y + 8 z = - 8$ .

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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 $8$ from both sides of the equation.

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

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 3.1.2

Subtract $8 z$ from both sides of the equation.

$4 y = - 8 - 8 - 8 z$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 3.1.3

Subtract $8$ from $- 8$ .

$4 y = - 16 - 8 z$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

$4 y = - 16 - 8 z$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 3.2

Divide each term in $4 y = - 16 - 8 z$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 y = - 16 - 8 z$ by $4$ .

$\frac{4 y}{4} = \frac{- 16}{4} + \frac{- 8 z}{4}$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{- 16}{4} + \frac{- 8 z}{4}$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 16}{4} + \frac{- 8 z}{4}$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

$y = \frac{- 16}{4} + \frac{- 8 z}{4}$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

$y = \frac{- 16}{4} + \frac{- 8 z}{4}$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

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 $- 16$ by $4$ .

$y = - 4 + \frac{- 8 z}{4}$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 3.2.3.1.2

Cancel the common factor of $- 8$ and $4$ .

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Step 3.2.3.1.2.1

Factor $4$ out of $- 8 z$ .

$y = - 4 + \frac{4 (- 2 z)}{4}$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 3.2.3.1.2.2

Cancel the common factors.

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Step 3.2.3.1.2.2.1

Factor $4$ out of $4$ .

$y = - 4 + \frac{4 (- 2 z)}{4 (1)}$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 3.2.3.1.2.2.2

Cancel the common factor.

$y = - 4 + \frac{4 (- 2 z)}{4 \cdot 1}$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 3.2.3.1.2.2.3

Rewrite the expression.

$y = - 4 + \frac{- 2 z}{1}$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 3.2.3.1.2.2.4

Divide $- 2 z$ by $1$ .

$y = - 4 - 2 z$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

$y = - 4 - 2 z$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

$y = - 4 - 2 z$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

$y = - 4 - 2 z$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

$y = - 4 - 2 z$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

$y = - 4 - 2 z$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

$y = - 4 - 2 z$

$4 - y - 2 z = 8$

$x = 2 + 3 y + z$

Step 4

Replace all occurrences of $y$ with $- 4 - 2 z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $4 - y - 2 z = 8$ with $- 4 - 2 z$ .

$4 - (- 4 - 2 z) - 2 z = 8$

$y = - 4 - 2 z$

$x = 2 + 3 y + z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $4 - (- 4 - 2 z) - 2 z$ .

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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.

$4 + 4 - (- 2 z) - 2 z = 8$

$y = - 4 - 2 z$

$x = 2 + 3 y + z$

Step 4.2.1.1.2

Multiply $- 1$ by $- 4$ .

$4 + 4 - (- 2 z) - 2 z = 8$

$y = - 4 - 2 z$

$x = 2 + 3 y + z$

Step 4.2.1.1.3

Multiply $- 2$ by $- 1$ .

$4 + 4 + 2 z - 2 z = 8$

$y = - 4 - 2 z$

$x = 2 + 3 y + z$

$4 + 4 + 2 z - 2 z = 8$

$y = - 4 - 2 z$

$x = 2 + 3 y + z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $4 + 4 + 2 z - 2 z$ .

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Step 4.2.1.2.1.1

Subtract $2 z$ from $2 z$ .

$4 + 4 + 0 = 8$

$y = - 4 - 2 z$

$x = 2 + 3 y + z$

Step 4.2.1.2.1.2

Add $4 + 4$ and $0$ .

$4 + 4 = 8$

$y = - 4 - 2 z$

$x = 2 + 3 y + z$

$4 + 4 = 8$

$y = - 4 - 2 z$

$x = 2 + 3 y + z$

Step 4.2.1.2.2

Add $4$ and $4$ .

$8 = 8$

$y = - 4 - 2 z$

$x = 2 + 3 y + z$

$8 = 8$

$y = - 4 - 2 z$

$x = 2 + 3 y + z$

$8 = 8$

$y = - 4 - 2 z$

$x = 2 + 3 y + z$

$8 = 8$

$y = - 4 - 2 z$

$x = 2 + 3 y + z$

Step 4.3

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

$x = 2 + 3 (- 4 - 2 z) + z$

$8 = 8$

$y = - 4 - 2 z$

Step 4.4

Simplify the right side.

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Step 4.4.1

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

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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.

$x = 2 + 3 \cdot - 4 + 3 (- 2 z) + z$

$8 = 8$

$y = - 4 - 2 z$

Step 4.4.1.1.2

Multiply $3$ by $- 4$ .

$x = 2 - 12 + 3 (- 2 z) + z$

$8 = 8$

$y = - 4 - 2 z$

Step 4.4.1.1.3

Multiply $- 2$ by $3$ .

$x = 2 - 12 - 6 z + z$

$8 = 8$

$y = - 4 - 2 z$

$x = 2 - 12 - 6 z + z$

$8 = 8$

$y = - 4 - 2 z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $12$ from $2$ .

$x = - 10 - 6 z + z$

$8 = 8$

$y = - 4 - 2 z$

Step 4.4.1.2.2

Add $- 6 z$ and $z$ .

$x = - 10 - 5 z$

$8 = 8$

$y = - 4 - 2 z$

$x = - 10 - 5 z$

$8 = 8$

$y = - 4 - 2 z$

$x = - 10 - 5 z$

$8 = 8$

$y = - 4 - 2 z$

$x = - 10 - 5 z$

$8 = 8$

$y = - 4 - 2 z$

$x = - 10 - 5 z$

$8 = 8$

$y = - 4 - 2 z$

Step 5

Remove any equations from the system that are always true.

$x = - 10 - 5 z$

$y = - 4 - 2 z$