Find Three Ordered Pair Solutions 3x+4y+6z=12

$3 x + 4 y + 6 z = 12$

Step 1

Solve the equation for $y$ .

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

Subtract $3 x$ from both sides of the equation.

$4 y + 6 z = 12 - 3 x$

Subtract $6 z$ from both sides of the equation.

$4 y = 12 - 3 x - 6 z$

Divide each term in $4 y = 12 - 3 x - 6 z$ by $4$ and simplify.

Divide each term in $4 y = 12 - 3 x - 6 z$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Divide $12$ by $4$ .

$y = 3 + \frac{- 3 x}{4} + \frac{- 6 z}{4}$

Move the negative in front of the fraction.

$y = 3 - \frac{(3) x}{4} + \frac{- 6 z}{4}$

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

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

$y = 3 - \frac{3 x}{4} + \frac{2 (- 3 z)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = 3 - \frac{3 x}{4} + \frac{2 (- 3 z)}{2 (2)}$

Cancel the common factor.

$y = 3 - \frac{3 x}{4} + \frac{2 (- 3 z)}{2 \cdot 2}$

Rewrite the expression.

$y = 3 - \frac{3 x}{4} + \frac{- 3 z}{2}$

Move the negative in front of the fraction.

$y = 3 - \frac{3 x}{4} - \frac{3 z}{2}$

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 $z$ .

Remove parentheses.

$y = 3 - \frac{3 (0)}{4} - \frac{3 (1)}{2}$

Simplify $3 - \frac{3 (0)}{4} - \frac{3 (1)}{2}$ .

Simplify each term.

Cancel the common factor of $0$ and $4$ .

Factor $4$ out of $3 (0)$ .

$y = 3 - \frac{4 (3 \cdot (0))}{4} - \frac{3 (1)}{2}$

Cancel the common factors.

Factor $4$ out of $4$ .

$y = 3 - \frac{4 (3 \cdot (0))}{4 (1)} - \frac{3 (1)}{2}$

Cancel the common factor.

$y = 3 - \frac{4 (3 \cdot (0))}{4 \cdot 1} - \frac{3 (1)}{2}$

Rewrite the expression.

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

Divide $3 \cdot (0)$ by $1$ .

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

Multiply $- (3 \cdot (0))$ .

Multiply $3$ by $0$ .

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

Multiply $- 1$ by $0$ .

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

Multiply $3$ by $1$ .

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

Add $3$ and $0$ .

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

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $3$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $2$ .

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

Subtract $3$ from $6$ .

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

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

$(0, \frac{3}{2}, 1)$

Step 4

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

Remove parentheses.

$y = 3 - \frac{3 (1)}{4} - \frac{3 (2)}{2}$

Simplify $3 - \frac{3 (1)}{4} - \frac{3 (2)}{2}$ .

Simplify each term.

Multiply $3$ by $1$ .

$y = 3 - \frac{3}{4} - \frac{3 (2)}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $3$ by $1$ .

$y = 3 - \frac{3}{4} - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$y = 3 - \frac{3}{4} - 3$

Find the common denominator.

Write $3$ as a fraction with denominator $1$ .

$y = \frac{3}{1} - \frac{3}{4} - 3$

Multiply $\frac{3}{1}$ by $\frac{4}{4}$ .

$y = \frac{3}{1} \cdot \frac{4}{4} - \frac{3}{4} - 3$

Multiply $\frac{3}{1}$ by $\frac{4}{4}$ .

$y = \frac{3 \cdot 4}{4} - \frac{3}{4} - 3$

Write $- 3$ as a fraction with denominator $1$ .

$y = \frac{3 \cdot 4}{4} - \frac{3}{4} + \frac{- 3}{1}$

Multiply $\frac{- 3}{1}$ by $\frac{4}{4}$ .

$y = \frac{3 \cdot 4}{4} - \frac{3}{4} + \frac{- 3}{1} \cdot \frac{4}{4}$

Multiply $\frac{- 3}{1}$ by $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

$y = \frac{3 \cdot 4 - 3 - 3 \cdot 4}{4}$

Simplify each term.

Multiply $3$ by $4$ .

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

Multiply $- 3$ by $4$ .

$y = \frac{12 - 3 - 12}{4}$

Simplify the expression.

Subtract $3$ from $12$ .

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

Subtract $12$ from $9$ .

$y = \frac{- 3}{4}$

Move the negative in front of the fraction.

$y = - \frac{3}{4}$

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

$(1, - \frac{3}{4}, 2)$

Step 5

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

Remove parentheses.

$y = 3 - \frac{3 (2)}{4} - \frac{3 (3)}{2}$

Simplify $3 - \frac{3 (2)}{4} - \frac{3 (3)}{2}$ .

Simplify each term.

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $3 (2)$ .

$y = 3 - \frac{2 \cdot 3}{4} - \frac{3 (3)}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $3$ .

$y = 3 - \frac{3}{2} - \frac{9}{2}$

Combine fractions.

Combine the numerators over the common denominator.

$y = 3 + \frac{- 3 - 9}{2}$

Simplify the expression.

Subtract $9$ from $- 3$ .

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

Divide $- 12$ by $2$ .

$y = 3 - 6$

Subtract $6$ from $3$ .

$y = - 3$

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

$(2, - 3, 3)$

Step 6

These are three possible solutions to the equation.

$(0, \frac{3}{2}, 1), (1, - \frac{3}{4}, 2), (2, - 3, 3)$

Find Three Ordered Pair Solutions 3x+4y+6z=12

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