Algebra Examples

$4 x - 6 y - 5 z = 60$

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 $4 x$ from both sides of the equation.

$- 6 y - 5 z = 60 - 4 x$

Step 1.1.2

Add $5 z$ to both sides of the equation.

$- 6 y = 60 - 4 x + 5 z$

Step 1.2

Divide each term in $- 6 y = 60 - 4 x + 5 z$ by $- 6$ and simplify.

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Step 1.2.1

Divide each term in $- 6 y = 60 - 4 x + 5 z$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{60}{- 6} + \frac{- 4 x}{- 6} + \frac{5 z}{- 6}$

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.

$\frac{- 6 y}{- 6} = \frac{60}{- 6} + \frac{- 4 x}{- 6} + \frac{5 z}{- 6}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{60}{- 6} + \frac{- 4 x}{- 6} + \frac{5 z}{- 6}$

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 $60$ by $- 6$ .

$y = - 10 + \frac{- 4 x}{- 6} + \frac{5 z}{- 6}$

Step 1.2.3.1.2

Cancel the common factor of $- 4$ and $- 6$ .

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Step 1.2.3.1.2.1

Factor $- 2$ out of $- 4 x$ .

$y = - 10 + \frac{- 2 (2 x)}{- 6} + \frac{5 z}{- 6}$

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 $- 6$ .

$y = - 10 + \frac{- 2 (2 x)}{- 2 (3)} + \frac{5 z}{- 6}$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y = - 10 + \frac{- 2 (2 x)}{- 2 \cdot 3} + \frac{5 z}{- 6}$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y = - 10 + \frac{2 x}{3} + \frac{5 z}{- 6}$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$y = - 10 + \frac{2 x}{3} - \frac{5 z}{6}$

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.

$y = - 10 + \frac{2 (0)}{3} - \frac{5 (1)}{6}$

Step 3.2

Simplify $- 10 + \frac{2 (0)}{3} - \frac{5 (1)}{6}$ .

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Step 3.2.1

Multiply $2$ by $0$ .

$y = - 10 + \frac{0}{3} - \frac{5 (1)}{6}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Divide $0$ by $3$ .

$y = - 10 + 0 - \frac{5 (1)}{6}$

Step 3.2.2.2

Multiply $5$ by $1$ .

$y = - 10 + 0 - \frac{5}{6}$

Step 3.2.3

Add $- 10$ and $0$ .

$y = - 10 - \frac{5}{6}$

Step 3.2.4

To write $- 10$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = - 10 \cdot \frac{6}{6} - \frac{5}{6}$

Step 3.2.5

Combine $- 10$ and $\frac{6}{6}$ .

$y = \frac{- 10 \cdot 6}{6} - \frac{5}{6}$

Step 3.2.6

Combine the numerators over the common denominator.

$y = \frac{- 10 \cdot 6 - 5}{6}$

Step 3.2.7

Simplify the numerator.

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Step 3.2.7.1

Multiply $- 10$ by $6$ .

$y = \frac{- 60 - 5}{6}$

Step 3.2.7.2

Subtract $5$ from $- 60$ .

$y = \frac{- 65}{6}$

Step 3.2.8

Move the negative in front of the fraction.

$y = - \frac{65}{6}$

Step 3.3

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

$(0, - \frac{65}{6}, 1)$

Step 4

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

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Step 4.1

Remove parentheses.

$y = - 10 + \frac{2 (1)}{3} - \frac{5 (2)}{6}$

Step 4.2

Simplify $- 10 + \frac{2 (1)}{3} - \frac{5 (2)}{6}$ .

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Step 4.2.1

Multiply $2$ by $1$ .

$y = - 10 + \frac{2}{3} - \frac{5 (2)}{6}$

Step 4.2.2

Find the common denominator.

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Step 4.2.2.1

Write $- 10$ as a fraction with denominator $1$ .

$y = \frac{- 10}{1} + \frac{2}{3} - \frac{5 (2)}{6}$

Step 4.2.2.2

Multiply $\frac{- 10}{1}$ by $\frac{6}{6}$ .

$y = \frac{- 10}{1} \cdot \frac{6}{6} + \frac{2}{3} - \frac{5 (2)}{6}$

Step 4.2.2.3

Multiply $\frac{- 10}{1}$ by $\frac{6}{6}$ .

$y = \frac{- 10 \cdot 6}{6} + \frac{2}{3} - \frac{5 (2)}{6}$

Step 4.2.2.4

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

$y = \frac{- 10 \cdot 6}{6} + \frac{2}{3} \cdot \frac{2}{2} - \frac{5 (2)}{6}$

Step 4.2.2.5

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

$y = \frac{- 10 \cdot 6}{6} + \frac{2 \cdot 2}{3 \cdot 2} - \frac{5 (2)}{6}$

Step 4.2.2.6

Reorder the factors of $3 \cdot 2$ .

$y = \frac{- 10 \cdot 6}{6} + \frac{2 \cdot 2}{2 \cdot 3} - \frac{5 (2)}{6}$

Step 4.2.2.7

Multiply $2$ by $3$ .

$y = \frac{- 10 \cdot 6}{6} + \frac{2 \cdot 2}{6} - \frac{5 (2)}{6}$

Step 4.2.3

Combine the numerators over the common denominator.

$y = \frac{- 10 \cdot 6 + 2 \cdot 2 - 5 \cdot 2}{6}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Multiply $- 10$ by $6$ .

$y = \frac{- 60 + 2 \cdot 2 - 5 \cdot 2}{6}$

Step 4.2.4.2

Multiply $2$ by $2$ .

$y = \frac{- 60 + 4 - 5 \cdot 2}{6}$

Step 4.2.4.3

Multiply $- 5$ by $2$ .

$y = \frac{- 60 + 4 - 10}{6}$

Step 4.2.5

Simplify the expression.

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Step 4.2.5.1

Add $- 60$ and $4$ .

$y = \frac{- 56 - 10}{6}$

Step 4.2.5.2

Subtract $10$ from $- 56$ .

$y = \frac{- 66}{6}$

Step 4.2.5.3

Divide $- 66$ by $6$ .

$y = - 11$

Step 4.3

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

$(1, - 11, 2)$

Step 5

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

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Step 5.1

Remove parentheses.

$y = - 10 + \frac{2 (2)}{3} - \frac{5 (3)}{6}$

Step 5.2

Simplify $- 10 + \frac{2 (2)}{3} - \frac{5 (3)}{6}$ .

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Step 5.2.1

Multiply $2$ by $2$ .

$y = - 10 + \frac{4}{3} - \frac{5 (3)}{6}$

Step 5.2.2

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

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Step 5.2.2.1

Factor $3$ out of $5 (3)$ .

$y = - 10 + \frac{4}{3} - \frac{3 \cdot 5}{6}$

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $3$ out of $6$ .

$y = - 10 + \frac{4}{3} - \frac{3 \cdot 5}{3 \cdot 2}$

Step 5.2.2.2.2

Cancel the common factor.

$y = - 10 + \frac{4}{3} - \frac{3 \cdot 5}{3 \cdot 2}$

Step 5.2.2.2.3

Rewrite the expression.

$y = - 10 + \frac{4}{3} - \frac{5}{2}$

Step 5.2.3

Find the common denominator.

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Step 5.2.3.1

Write $- 10$ as a fraction with denominator $1$ .

$y = \frac{- 10}{1} + \frac{4}{3} - \frac{5}{2}$

Step 5.2.3.2

Multiply $\frac{- 10}{1}$ by $\frac{6}{6}$ .

$y = \frac{- 10}{1} \cdot \frac{6}{6} + \frac{4}{3} - \frac{5}{2}$

Step 5.2.3.3

Multiply $\frac{- 10}{1}$ by $\frac{6}{6}$ .

$y = \frac{- 10 \cdot 6}{6} + \frac{4}{3} - \frac{5}{2}$

Step 5.2.3.4

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$y = \frac{- 10 \cdot 6}{6} + \frac{4}{3} \cdot \frac{2}{2} - \frac{5}{2}$

Step 5.2.3.5

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$y = \frac{- 10 \cdot 6}{6} + \frac{4 \cdot 2}{3 \cdot 2} - \frac{5}{2}$

Step 5.2.3.6

Multiply $\frac{5}{2}$ by $\frac{3}{3}$ .

$y = \frac{- 10 \cdot 6}{6} + \frac{4 \cdot 2}{3 \cdot 2} - (\frac{5}{2} \cdot \frac{3}{3})$

Step 5.2.3.7

Multiply $\frac{5}{2}$ by $\frac{3}{3}$ .

$y = \frac{- 10 \cdot 6}{6} + \frac{4 \cdot 2}{3 \cdot 2} - \frac{5 \cdot 3}{2 \cdot 3}$

Step 5.2.3.8

Reorder the factors of $3 \cdot 2$ .

$y = \frac{- 10 \cdot 6}{6} + \frac{4 \cdot 2}{2 \cdot 3} - \frac{5 \cdot 3}{2 \cdot 3}$

Step 5.2.3.9

Multiply $2$ by $3$ .

$y = \frac{- 10 \cdot 6}{6} + \frac{4 \cdot 2}{6} - \frac{5 \cdot 3}{2 \cdot 3}$

Step 5.2.3.10

Multiply $2$ by $3$ .

$y = \frac{- 10 \cdot 6}{6} + \frac{4 \cdot 2}{6} - \frac{5 \cdot 3}{6}$

Step 5.2.4

Combine the numerators over the common denominator.

$y = \frac{- 10 \cdot 6 + 4 \cdot 2 - 5 \cdot 3}{6}$

Step 5.2.5

Simplify each term.

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Step 5.2.5.1

Multiply $- 10$ by $6$ .

$y = \frac{- 60 + 4 \cdot 2 - 5 \cdot 3}{6}$

Step 5.2.5.2

Multiply $4$ by $2$ .

$y = \frac{- 60 + 8 - 5 \cdot 3}{6}$

Step 5.2.5.3

Multiply $- 5$ by $3$ .

$y = \frac{- 60 + 8 - 15}{6}$

Step 5.2.6

Simplify the expression.

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Step 5.2.6.1

Add $- 60$ and $8$ .

$y = \frac{- 52 - 15}{6}$

Step 5.2.6.2

Subtract $15$ from $- 52$ .

$y = \frac{- 67}{6}$

Step 5.2.6.3

Move the negative in front of the fraction.

$y = - \frac{67}{6}$

Step 5.3

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

$(2, - \frac{67}{6}, 3)$

Step 6

These are three possible solutions to the equation.

$(0, - \frac{65}{6}, 1), (1, - 11, 2), (2, - \frac{67}{6}, 3)$