Algebra Examples

$- 2 x + y + 2 z = 12$ $2 x - 4 y + z = - 15$ $y + 4 z = 6$

Step 1

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

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Step 1.1

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

$y + 2 z = 12 + 2 x$

$2 x - 4 y + z = - 15$

$y + 4 z = 6$

Step 1.2

Subtract $2 z$ from both sides of the equation.

$y = 12 + 2 x - 2 z$

$2 x - 4 y + z = - 15$

$y + 4 z = 6$

$y = 12 + 2 x - 2 z$

$2 x - 4 y + z = - 15$

$y + 4 z = 6$

Step 2

Replace all occurrences of $y$ with $12 + 2 x - 2 z$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $2 x - 4 y + z = - 15$ with $12 + 2 x - 2 z$ .

$2 x - 4 (12 + 2 x - 2 z) + z = - 15$

$y = 12 + 2 x - 2 z$

$y + 4 z = 6$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 x - 4 (12 + 2 x - 2 z) + z$ .

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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.

$2 x - 4 \cdot 12 - 4 (2 x) - 4 (- 2 z) + z = - 15$

$y = 12 + 2 x - 2 z$

$y + 4 z = 6$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 4$ by $12$ .

$2 x - 48 - 4 (2 x) - 4 (- 2 z) + z = - 15$

$y = 12 + 2 x - 2 z$

$y + 4 z = 6$

Step 2.2.1.1.2.2

Multiply $2$ by $- 4$ .

$2 x - 48 - 8 x - 4 (- 2 z) + z = - 15$

$y = 12 + 2 x - 2 z$

$y + 4 z = 6$

Step 2.2.1.1.2.3

Multiply $- 2$ by $- 4$ .

$2 x - 48 - 8 x + 8 z + z = - 15$

$y = 12 + 2 x - 2 z$

$y + 4 z = 6$

$2 x - 48 - 8 x + 8 z + z = - 15$

$y = 12 + 2 x - 2 z$

$y + 4 z = 6$

$2 x - 48 - 8 x + 8 z + z = - 15$

$y = 12 + 2 x - 2 z$

$y + 4 z = 6$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $8 x$ from $2 x$ .

$- 6 x - 48 + 8 z + z = - 15$

$y = 12 + 2 x - 2 z$

$y + 4 z = 6$

Step 2.2.1.2.2

Add $8 z$ and $z$ .

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$y + 4 z = 6$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$y + 4 z = 6$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$y + 4 z = 6$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$y + 4 z = 6$

Step 2.3

Replace all occurrences of $y$ in $y + 4 z = 6$ with $12 + 2 x - 2 z$ .

$(12 + 2 x - 2 z) + 4 z = 6$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Add $- 2 z$ and $4 z$ .

$12 + 2 x + 2 z = 6$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$12 + 2 x + 2 z = 6$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$12 + 2 x + 2 z = 6$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

Step 3

Solve for $x$ in $12 + 2 x + 2 z = 6$ .

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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

Subtract $12$ from both sides of the equation.

$2 x + 2 z = 6 - 12$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

Step 3.1.2

Subtract $2 z$ from both sides of the equation.

$2 x = 6 - 12 - 2 z$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

Step 3.1.3

Subtract $12$ from $6$ .

$2 x = - 6 - 2 z$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$2 x = - 6 - 2 z$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

Step 3.2

Divide each term in $2 x = - 6 - 2 z$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x = - 6 - 2 z$ by $2$ .

$\frac{2 x}{2} = \frac{- 6}{2} + \frac{- 2 z}{2}$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 6}{2} + \frac{- 2 z}{2}$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{2} + \frac{- 2 z}{2}$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$x = \frac{- 6}{2} + \frac{- 2 z}{2}$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$x = \frac{- 6}{2} + \frac{- 2 z}{2}$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

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 $- 6$ by $2$ .

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

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

Step 3.2.3.1.2

Cancel the common factor of $- 2$ and $2$ .

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Step 3.2.3.1.2.1

Factor $2$ out of $- 2 z$ .

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

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

Step 3.2.3.1.2.2

Cancel the common factors.

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Step 3.2.3.1.2.2.1

Factor $2$ out of $2$ .

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

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

Step 3.2.3.1.2.2.2

Cancel the common factor.

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

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

Step 3.2.3.1.2.2.3

Rewrite the expression.

$x = - 3 + \frac{- z}{1}$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

Step 3.2.3.1.2.2.4

Divide $- z$ by $1$ .

$x = - 3 - z$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$x = - 3 - z$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$x = - 3 - z$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$x = - 3 - z$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$x = - 3 - z$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$x = - 3 - z$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

$x = - 3 - z$

$- 6 x - 48 + 9 z = - 15$

$y = 12 + 2 x - 2 z$

Step 4

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

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Step 4.1

Replace all occurrences of $x$ in $- 6 x - 48 + 9 z = - 15$ with $- 3 - z$ .

$- 6 (- 3 - z) - 48 + 9 z = - 15$

$x = - 3 - z$

$y = 12 + 2 x - 2 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 6 (- 3 - z) - 48 + 9 z$ .

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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.

$- 6 \cdot - 3 - 6 (- z) - 48 + 9 z = - 15$

$x = - 3 - z$

$y = 12 + 2 x - 2 z$

Step 4.2.1.1.2

Multiply $- 6$ by $- 3$ .

$18 - 6 (- z) - 48 + 9 z = - 15$

$x = - 3 - z$

$y = 12 + 2 x - 2 z$

Step 4.2.1.1.3

Multiply $- 1$ by $- 6$ .

$18 + 6 z - 48 + 9 z = - 15$

$x = - 3 - z$

$y = 12 + 2 x - 2 z$

$18 + 6 z - 48 + 9 z = - 15$

$x = - 3 - z$

$y = 12 + 2 x - 2 z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $48$ from $18$ .

$6 z - 30 + 9 z = - 15$

$x = - 3 - z$

$y = 12 + 2 x - 2 z$

Step 4.2.1.2.2

Add $6 z$ and $9 z$ .

$15 z - 30 = - 15$

$x = - 3 - z$

$y = 12 + 2 x - 2 z$

$15 z - 30 = - 15$

$x = - 3 - z$

$y = 12 + 2 x - 2 z$

$15 z - 30 = - 15$

$x = - 3 - z$

$y = 12 + 2 x - 2 z$

$15 z - 30 = - 15$

$x = - 3 - z$

$y = 12 + 2 x - 2 z$

Step 4.3

Replace all occurrences of $x$ in $y = 12 + 2 x - 2 z$ with $- 3 - z$ .

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

$15 z - 30 = - 15$

$x = - 3 - z$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $12 + 2 (- 3 - z) - 2 z$ .

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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.

$y = 12 + 2 \cdot - 3 + 2 (- z) - 2 z$

$15 z - 30 = - 15$

$x = - 3 - z$

Step 4.4.1.1.2

Multiply $2$ by $- 3$ .

$y = 12 - 6 + 2 (- z) - 2 z$

$15 z - 30 = - 15$

$x = - 3 - z$

Step 4.4.1.1.3

Multiply $- 1$ by $2$ .

$y = 12 - 6 - 2 z - 2 z$

$15 z - 30 = - 15$

$x = - 3 - z$

$y = 12 - 6 - 2 z - 2 z$

$15 z - 30 = - 15$

$x = - 3 - z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $6$ from $12$ .

$y = 6 - 2 z - 2 z$

$15 z - 30 = - 15$

$x = - 3 - z$

Step 4.4.1.2.2

Subtract $2 z$ from $- 2 z$ .

$y = 6 - 4 z$

$15 z - 30 = - 15$

$x = - 3 - z$

$y = 6 - 4 z$

$15 z - 30 = - 15$

$x = - 3 - z$

$y = 6 - 4 z$

$15 z - 30 = - 15$

$x = - 3 - z$

$y = 6 - 4 z$

$15 z - 30 = - 15$

$x = - 3 - z$

$y = 6 - 4 z$

$15 z - 30 = - 15$

$x = - 3 - z$

Step 5

Solve for $z$ in $15 z - 30 = - 15$ .

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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 $30$ to both sides of the equation.

$15 z = - 15 + 30$

$y = 6 - 4 z$

$x = - 3 - z$

Step 5.1.2

Add $- 15$ and $30$ .

$15 z = 15$

$y = 6 - 4 z$

$x = - 3 - z$

$15 z = 15$

$y = 6 - 4 z$

$x = - 3 - z$

Step 5.2

Divide each term in $15 z = 15$ by $15$ and simplify.

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Step 5.2.1

Divide each term in $15 z = 15$ by $15$ .

$\frac{15 z}{15} = \frac{15}{15}$

$y = 6 - 4 z$

$x = - 3 - z$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $15$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{15 z}{15} = \frac{15}{15}$

$y = 6 - 4 z$

$x = - 3 - z$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{15}{15}$

$y = 6 - 4 z$

$x = - 3 - z$

$z = \frac{15}{15}$

$y = 6 - 4 z$

$x = - 3 - z$

$z = \frac{15}{15}$

$y = 6 - 4 z$

$x = - 3 - z$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $15$ by $15$ .

$z = 1$

$y = 6 - 4 z$

$x = - 3 - z$

$z = 1$

$y = 6 - 4 z$

$x = - 3 - z$

$z = 1$

$y = 6 - 4 z$

$x = - 3 - z$

$z = 1$

$y = 6 - 4 z$

$x = - 3 - z$

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 $y = 6 - 4 z$ with $1$ .

$y = 6 - 4 \cdot 1$

$z = 1$

$x = - 3 - z$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $6 - 4 \cdot 1$ .

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Step 6.2.1.1

Multiply $- 4$ by $1$ .

$y = 6 - 4$

$z = 1$

$x = - 3 - z$

Step 6.2.1.2

Subtract $4$ from $6$ .

$y = 2$

$z = 1$

$x = - 3 - z$

$y = 2$

$z = 1$

$x = - 3 - z$

$y = 2$

$z = 1$

$x = - 3 - z$

Step 6.3

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

$x = - 3 - (1)$

$y = 2$

$z = 1$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- 3 - (1)$ .

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Step 6.4.1.1

Multiply $- 1$ by $1$ .

$x = - 3 - 1$

$y = 2$

$z = 1$

Step 6.4.1.2

Subtract $1$ from $- 3$ .

$x = - 4$

$y = 2$

$z = 1$

$x = - 4$

$y = 2$

$z = 1$

$x = - 4$

$y = 2$

$z = 1$

$x = - 4$

$y = 2$

$z = 1$

Step 7

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

$(- 4, 2, 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 4, 2, 1)$

Equation Form:

$x = - 4, y = 2, z = 1$