Find Three Ordered Pair Solutions 4x+6y+2z=12

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

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.

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

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

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

Divide each term in $6 y = 12 - 4 x - 2 z$ by $6$ and simplify.

Divide each term in $6 y = 12 - 4 x - 2 z$ by $6$ .

$\frac{6 y}{6} = \frac{12}{6} + \frac{- 4 x}{6} + \frac{- 2 z}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{12}{6} + \frac{- 4 x}{6} + \frac{- 2 z}{6}$

Divide $y$ by $1$ .

$y = \frac{12}{6} + \frac{- 4 x}{6} + \frac{- 2 z}{6}$

Simplify the right side.

Simplify each term.

Divide $12$ by $6$ .

$y = 2 + \frac{- 4 x}{6} + \frac{- 2 z}{6}$

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

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

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

Cancel the common factors.

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

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

Cancel the common factors.

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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.

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

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

Combine the numerators over the common denominator.

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

Simplify the expression.

Multiply $- 2$ by $0$ .

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

Subtract $1$ from $0$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

$y = \frac{6 - 1}{3}$

Subtract $1$ from $6$ .

$y = \frac{5}{3}$

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

$(0, \frac{5}{3}, 1)$

Step 4

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

Remove parentheses.

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

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

Combine the numerators over the common denominator.

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

Simplify the expression.

Multiply $- 2$ by $1$ .

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

Subtract $2$ from $- 2$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

$y = \frac{6 - 4}{3}$

Subtract $4$ from $6$ .

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

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

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

Step 5

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

Remove parentheses.

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

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

Combine the numerators over the common denominator.

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

Simplify the expression.

Multiply $- 2$ by $2$ .

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

Subtract $3$ from $- 4$ .

$y = 2 + \frac{- 7}{3}$

Move the negative in front of the fraction.

$y = 2 - \frac{7}{3}$

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

$y = 2 \cdot \frac{3}{3} - \frac{7}{3}$

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

$y = \frac{2 \cdot 3}{3} - \frac{7}{3}$

Combine the numerators over the common denominator.

$y = \frac{2 \cdot 3 - 7}{3}$

Simplify the numerator.

Multiply $2$ by $3$ .

$y = \frac{6 - 7}{3}$

Subtract $7$ from $6$ .

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

Move the negative in front of the fraction.

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

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

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

Step 6

These are three possible solutions to the equation.

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

Find Three Ordered Pair Solutions 4x+6y+2z=12

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