Find Three Ordered Pair Solutions 4x-6y-5z=60

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

Step 1

Solve the equation for $y$ .

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

Subtract $4 x$ from both sides of the equation.

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

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

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

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

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

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

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Divide $60$ by $- 6$ .

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

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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 $z$ .

Remove parentheses.

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

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

Multiply $2$ by $0$ .

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

Simplify each term.

Divide $0$ by $3$ .

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

Multiply $5$ by $1$ .

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

Add $- 10$ and $0$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 10$ by $6$ .

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

Subtract $5$ from $- 60$ .

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

Move the negative in front of the fraction.

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

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 $z$ .

Remove parentheses.

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

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

Multiply $2$ by $1$ .

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

Find the common denominator.

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

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

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

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

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

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

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

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

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

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

Reorder the factors of $3 \cdot 2$ .

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

Multiply $2$ by $3$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Multiply $- 10$ by $6$ .

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

Multiply $2$ by $2$ .

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

Multiply $- 5$ by $2$ .

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

Simplify the expression.

Add $- 60$ and $4$ .

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

Subtract $10$ from $- 56$ .

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

Divide $- 66$ by $6$ .

$y = - 11$

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 $z$ .

Remove parentheses.

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

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

Multiply $2$ by $2$ .

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

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

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

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

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Find the common denominator.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Reorder the factors of $3 \cdot 2$ .

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

Multiply $2$ by $3$ .

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

Multiply $2$ by $3$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Multiply $- 10$ by $6$ .

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

Multiply $4$ by $2$ .

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

Multiply $- 5$ by $3$ .

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

Simplify the expression.

Add $- 60$ and $8$ .

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

Subtract $15$ from $- 52$ .

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

Move the negative in front of the fraction.

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

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.

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

Find Three Ordered Pair Solutions 4x-6y-5z=60

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