Algebra Examples

$2 x - 6 y - 3 z = 12$

Step 1

Solve the equation for $y$ .

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Step 1.1

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

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Step 1.1.1

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

$- 6 y - 3 z = 12 - 2 x$

Step 1.1.2

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

$- 6 y = 12 - 2 x + 3 z$

Step 1.2

Divide each term in $- 6 y = 12 - 2 x + 3 z$ by $- 6$ and simplify.

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Step 1.2.1

Divide each term in $- 6 y = 12 - 2 x + 3 z$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{12}{- 6} + \frac{- 2 x}{- 6} + \frac{3 z}{- 6}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 6$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{12}{- 6} + \frac{- 2 x}{- 6} + \frac{3 z}{- 6}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{12}{- 6} + \frac{- 2 x}{- 6} + \frac{3 z}{- 6}$

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 $12$ by $- 6$ .

$y = - 2 + \frac{- 2 x}{- 6} + \frac{3 z}{- 6}$

Step 1.2.3.1.2

Cancel the common factor of $- 2$ and $- 6$ .

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Step 1.2.3.1.2.1

Factor $- 2$ out of $- 2 x$ .

$y = - 2 + \frac{- 2 (x)}{- 6} + \frac{3 z}{- 6}$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $- 2$ out of $- 6$ .

$y = - 2 + \frac{- 2 (x)}{- 2 (3)} + \frac{3 z}{- 6}$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y = - 2 + \frac{- 2 x}{- 2 \cdot 3} + \frac{3 z}{- 6}$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y = - 2 + \frac{x}{3} + \frac{3 z}{- 6}$

Step 1.2.3.1.3

Cancel the common factor of $3$ and $- 6$ .

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Step 1.2.3.1.3.1

Factor $3$ out of $3 z$ .

$y = - 2 + \frac{x}{3} + \frac{3 (z)}{- 6}$

Step 1.2.3.1.3.2

Cancel the common factors.

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Step 1.2.3.1.3.2.1

Factor $3$ out of $- 6$ .

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

Step 1.2.3.1.3.2.2

Cancel the common factor.

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

Step 1.2.3.1.3.2.3

Rewrite the expression.

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

Step 1.2.3.1.4

Move the negative in front of the fraction.

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

Step 2

Choose any values for $x$ and $y$ that are in the domain to plug into the equation.

Step 3

Choose $0$ to substitute in for $x$ and $1$ to substitute for $z$ .

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Step 3.1

Remove parentheses.

$y = - 2 + \frac{0}{3} - \frac{1}{2}$

Step 3.2

Simplify $- 2 + \frac{0}{3} - \frac{1}{2}$ .

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Step 3.2.1

Find the common denominator.

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Step 3.2.1.1

Write $- 2$ as a fraction with denominator $1$ .

$y = \frac{- 2}{1} + \frac{0}{3} - \frac{1}{2}$

Step 3.2.1.2

Multiply $\frac{- 2}{1}$ by $\frac{6}{6}$ .

$y = \frac{- 2}{1} \cdot \frac{6}{6} + \frac{0}{3} - \frac{1}{2}$

Step 3.2.1.3

Multiply $\frac{- 2}{1}$ by $\frac{6}{6}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{0}{3} - \frac{1}{2}$

Step 3.2.1.4

Multiply $\frac{0}{3}$ by $\frac{2}{2}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{0}{3} \cdot \frac{2}{2} - \frac{1}{2}$

Step 3.2.1.5

Multiply $\frac{0}{3}$ by $\frac{2}{2}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{0 \cdot 2}{3 \cdot 2} - \frac{1}{2}$

Step 3.2.1.6

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{0 \cdot 2}{3 \cdot 2} - (\frac{1}{2} \cdot \frac{3}{3})$

Step 3.2.1.7

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{0 \cdot 2}{3 \cdot 2} - \frac{3}{2 \cdot 3}$

Step 3.2.1.8

Reorder the factors of $3 \cdot 2$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{0 \cdot 2}{2 \cdot 3} - \frac{3}{2 \cdot 3}$

Step 3.2.1.9

Multiply $2$ by $3$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{0 \cdot 2}{6} - \frac{3}{2 \cdot 3}$

Step 3.2.1.10

Multiply $2$ by $3$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{0 \cdot 2}{6} - \frac{3}{6}$

Step 3.2.2

Combine the numerators over the common denominator.

$y = \frac{- 2 \cdot 6 + 0 \cdot 2 - 3}{6}$

Step 3.2.3

Simplify each term.

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Step 3.2.3.1

Multiply $- 2$ by $6$ .

$y = \frac{- 12 + 0 \cdot 2 - 3}{6}$

Step 3.2.3.2

Multiply $0$ by $2$ .

$y = \frac{- 12 + 0 - 3}{6}$

Step 3.2.4

Reduce the expression by cancelling the common factors.

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Step 3.2.4.1

Add $- 12$ and $0$ .

$y = \frac{- 12 - 3}{6}$

Step 3.2.4.2

Subtract $3$ from $- 12$ .

$y = \frac{- 15}{6}$

Step 3.2.4.3

Cancel the common factor of $- 15$ and $6$ .

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Step 3.2.4.3.1

Factor $3$ out of $- 15$ .

$y = \frac{3 (- 5)}{6}$

Step 3.2.4.3.2

Cancel the common factors.

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Step 3.2.4.3.2.1

Factor $3$ out of $6$ .

$y = \frac{3 \cdot - 5}{3 \cdot 2}$

Step 3.2.4.3.2.2

Cancel the common factor.

$y = \frac{3 \cdot - 5}{3 \cdot 2}$

Step 3.2.4.3.2.3

Rewrite the expression.

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

Step 3.2.4.4

Move the negative in front of the fraction.

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

Step 3.3

Use the $x$ , $y$ , and $z$ values to form the ordered pair.

$(0, - \frac{5}{2}, 1)$

Step 4

Choose $1$ to substitute in for $x$ and $2$ to substitute for $z$ .

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Step 4.1

Remove parentheses.

$y = - 2 + \frac{1}{3} - \frac{2}{2}$

Step 4.2

Simplify $- 2 + \frac{1}{3} - \frac{2}{2}$ .

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Step 4.2.1

Find the common denominator.

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Step 4.2.1.1

Write $- 2$ as a fraction with denominator $1$ .

$y = \frac{- 2}{1} + \frac{1}{3} - \frac{2}{2}$

Step 4.2.1.2

Multiply $\frac{- 2}{1}$ by $\frac{6}{6}$ .

$y = \frac{- 2}{1} \cdot \frac{6}{6} + \frac{1}{3} - \frac{2}{2}$

Step 4.2.1.3

Multiply $\frac{- 2}{1}$ by $\frac{6}{6}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{1}{3} - \frac{2}{2}$

Step 4.2.1.4

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{1}{3} \cdot \frac{2}{2} - \frac{2}{2}$

Step 4.2.1.5

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{2}{3 \cdot 2} - \frac{2}{2}$

Step 4.2.1.6

Multiply $\frac{2}{2}$ by $\frac{3}{3}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{2}{3 \cdot 2} - (\frac{2}{2} \cdot \frac{3}{3})$

Step 4.2.1.7

Multiply $\frac{2}{2}$ by $\frac{3}{3}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{2}{3 \cdot 2} - \frac{2 \cdot 3}{2 \cdot 3}$

Step 4.2.1.8

Reorder the factors of $3 \cdot 2$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{2}{2 \cdot 3} - \frac{2 \cdot 3}{2 \cdot 3}$

Step 4.2.1.9

Multiply $2$ by $3$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{2}{6} - \frac{2 \cdot 3}{2 \cdot 3}$

Step 4.2.1.10

Multiply $2$ by $3$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{2}{6} - \frac{2 \cdot 3}{6}$

Step 4.2.2

Combine the numerators over the common denominator.

$y = \frac{- 2 \cdot 6 + 2 - 2 \cdot 3}{6}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Multiply $- 2$ by $6$ .

$y = \frac{- 12 + 2 - 2 \cdot 3}{6}$

Step 4.2.3.2

Multiply $- 2$ by $3$ .

$y = \frac{- 12 + 2 - 6}{6}$

Step 4.2.4

Reduce the expression by cancelling the common factors.

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Step 4.2.4.1

Add $- 12$ and $2$ .

$y = \frac{- 10 - 6}{6}$

Step 4.2.4.2

Subtract $6$ from $- 10$ .

$y = \frac{- 16}{6}$

Step 4.2.4.3

Cancel the common factor of $- 16$ and $6$ .

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Step 4.2.4.3.1

Factor $2$ out of $- 16$ .

$y = \frac{2 (- 8)}{6}$

Step 4.2.4.3.2

Cancel the common factors.

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Step 4.2.4.3.2.1

Factor $2$ out of $6$ .

$y = \frac{2 \cdot - 8}{2 \cdot 3}$

Step 4.2.4.3.2.2

Cancel the common factor.

$y = \frac{2 \cdot - 8}{2 \cdot 3}$

Step 4.2.4.3.2.3

Rewrite the expression.

$y = \frac{- 8}{3}$

Step 4.2.4.4

Move the negative in front of the fraction.

$y = - \frac{8}{3}$

Step 4.3

Use the $x$ , $y$ , and $z$ values to form the ordered pair.

$(1, - \frac{8}{3}, 2)$

Step 5

Choose $2$ to substitute in for $x$ and $3$ to substitute for $z$ .

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Step 5.1

Remove parentheses.

$y = - 2 + \frac{2}{3} - \frac{3}{2}$

Step 5.2

Simplify $- 2 + \frac{2}{3} - \frac{3}{2}$ .

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Step 5.2.1

Find the common denominator.

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Step 5.2.1.1

Write $- 2$ as a fraction with denominator $1$ .

$y = \frac{- 2}{1} + \frac{2}{3} - \frac{3}{2}$

Step 5.2.1.2

Multiply $\frac{- 2}{1}$ by $\frac{6}{6}$ .

$y = \frac{- 2}{1} \cdot \frac{6}{6} + \frac{2}{3} - \frac{3}{2}$

Step 5.2.1.3

Multiply $\frac{- 2}{1}$ by $\frac{6}{6}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{2}{3} - \frac{3}{2}$

Step 5.2.1.4

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{2}{3} \cdot \frac{2}{2} - \frac{3}{2}$

Step 5.2.1.5

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{2 \cdot 2}{3 \cdot 2} - \frac{3}{2}$

Step 5.2.1.6

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{2 \cdot 2}{3 \cdot 2} - (\frac{3}{2} \cdot \frac{3}{3})$

Step 5.2.1.7

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{2 \cdot 2}{3 \cdot 2} - \frac{3 \cdot 3}{2 \cdot 3}$

Step 5.2.1.8

Reorder the factors of $3 \cdot 2$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{2 \cdot 2}{2 \cdot 3} - \frac{3 \cdot 3}{2 \cdot 3}$

Step 5.2.1.9

Multiply $2$ by $3$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{2 \cdot 2}{6} - \frac{3 \cdot 3}{2 \cdot 3}$

Step 5.2.1.10

Multiply $2$ by $3$ .

$y = \frac{- 2 \cdot 6}{6} + \frac{2 \cdot 2}{6} - \frac{3 \cdot 3}{6}$

Step 5.2.2

Combine the numerators over the common denominator.

$y = \frac{- 2 \cdot 6 + 2 \cdot 2 - 3 \cdot 3}{6}$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Multiply $- 2$ by $6$ .

$y = \frac{- 12 + 2 \cdot 2 - 3 \cdot 3}{6}$

Step 5.2.3.2

Multiply $2$ by $2$ .

$y = \frac{- 12 + 4 - 3 \cdot 3}{6}$

Step 5.2.3.3

Multiply $- 3$ by $3$ .

$y = \frac{- 12 + 4 - 9}{6}$

Step 5.2.4

Simplify the expression.

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Step 5.2.4.1

Add $- 12$ and $4$ .

$y = \frac{- 8 - 9}{6}$

Step 5.2.4.2

Subtract $9$ from $- 8$ .

$y = \frac{- 17}{6}$

Step 5.2.4.3

Move the negative in front of the fraction.

$y = - \frac{17}{6}$

Step 5.3

Use the $x$ , $y$ , and $z$ values to form the ordered pair.

$(2, - \frac{17}{6}, 3)$

Step 6

These are three possible solutions to the equation.

$(0, - \frac{5}{2}, 1), (1, - \frac{8}{3}, 2), (2, - \frac{17}{6}, 3)$