Find Three Ordered Pair Solutions x+3y+2z=6

$x + 3 y + 2 z = 6$

Step 1

Solve the equation for $y$ .

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

Subtract $x$ from both sides of the equation.

$3 y + 2 z = 6 - x$

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

$3 y = 6 - x - 2 z$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Divide $6$ by $3$ .

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

Move the negative in front of the fraction.

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

Move the negative in front of the fraction.

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

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.

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

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

Simplify each term.

Divide $0$ by $3$ .

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

Multiply $- 1$ by $0$ .

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

Multiply $2$ by $1$ .

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

Add $2$ and $0$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

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

Subtract $2$ from $6$ .

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

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

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

Step 4

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

Remove parentheses.

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

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

Combine the numerators over the common denominator.

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

Simplify the expression.

Multiply $- 2$ by $2$ .

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

Subtract $4$ from $- 1$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

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

Subtract $5$ from $6$ .

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

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

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

Step 5

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

Remove parentheses.

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

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

Combine the numerators over the common denominator.

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

Simplify the expression.

Multiply $- 2$ by $3$ .

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

Subtract $6$ from $- 2$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

$y = \frac{6 - 8}{3}$

Subtract $8$ from $6$ .

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

Move the negative in front of the fraction.

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

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

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

Step 6

These are three possible solutions to the equation.

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

Find Three Ordered Pair Solutions x+3y+2z=6

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