Algebra Examples

$x + 2 y + 3 z = 6$

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 $x$ from both sides of the equation.

$2 y + 3 z = 6 - x$

Step 1.1.2

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

$2 y = 6 - x - 3 z$

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide $y$ by $1$ .

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

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 $6$ by $2$ .

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

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

Step 1.2.3.1.3

Move the negative in front of the fraction.

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

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 $z$ .

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Step 3.1

Remove parentheses.

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

Step 3.2

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

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Divide $0$ by $2$ .

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

Step 3.2.1.2

Multiply $- 1$ by $0$ .

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

Step 3.2.1.3

Multiply $3$ by $1$ .

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

Step 3.2.2

Add $3$ and $0$ .

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Combine the numerators over the common denominator.

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

Step 3.2.6

Simplify the numerator.

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Step 3.2.6.1

Multiply $3$ by $2$ .

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

Step 3.2.6.2

Subtract $3$ from $6$ .

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

Step 3.3

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 $z$ .

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Step 4.1

Remove parentheses.

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

Step 4.2

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

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Step 4.2.1

Combine the numerators over the common denominator.

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

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Multiply $- 3$ by $2$ .

$y = 3 + \frac{- 1 - 6}{2}$

Step 4.2.2.2

Subtract $6$ from $- 1$ .

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

Step 4.2.2.3

Move the negative in front of the fraction.

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

Combine the numerators over the common denominator.

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

Step 4.2.6

Simplify the numerator.

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Step 4.2.6.1

Multiply $3$ by $2$ .

$y = \frac{6 - 7}{2}$

Step 4.2.6.2

Subtract $7$ from $6$ .

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

Step 4.2.7

Move the negative in front of the fraction.

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

Step 4.3

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

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

Step 5

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

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Step 5.1

Remove parentheses.

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

Step 5.2

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

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Step 5.2.1

Combine the numerators over the common denominator.

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

Step 5.2.2

Simplify the expression.

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Step 5.2.2.1

Multiply $- 3$ by $3$ .

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

Step 5.2.2.2

Subtract $9$ from $- 2$ .

$y = 3 + \frac{- 11}{2}$

Step 5.2.2.3

Move the negative in front of the fraction.

$y = 3 - \frac{11}{2}$

Step 5.2.3

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

$y = 3 \cdot \frac{2}{2} - \frac{11}{2}$

Step 5.2.4

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

$y = \frac{3 \cdot 2}{2} - \frac{11}{2}$

Step 5.2.5

Combine the numerators over the common denominator.

$y = \frac{3 \cdot 2 - 11}{2}$

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

Multiply $3$ by $2$ .

$y = \frac{6 - 11}{2}$

Step 5.2.6.2

Subtract $11$ from $6$ .

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

Step 5.2.7

Move the negative in front of the fraction.

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

Step 5.3

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

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

Step 6

These are three possible solutions to the equation.

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