Algebra Examples

$x - 5 y - z = - 10$

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.

$- 5 y - z = - 10 - x$

Step 1.1.2

Add $z$ to both sides of the equation.

$- 5 y = - 10 - x + z$

Step 1.2

Divide each term in $- 5 y = - 10 - x + z$ by $- 5$ and simplify.

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Step 1.2.1

Divide each term in $- 5 y = - 10 - x + z$ by $- 5$ .

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 5$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide $y$ by $1$ .

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

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 $- 10$ by $- 5$ .

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

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

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

Step 1.2.3.1.3

Move the negative in front of the fraction.

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

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.

$y = 2 + \frac{0}{5} - \frac{1}{5}$

Step 3.2

Simplify $2 + \frac{0}{5} - \frac{1}{5}$ .

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Step 3.2.1

Find the common denominator.

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Step 3.2.1.1

Write $2$ as a fraction with denominator $1$ .

$y = \frac{2}{1} + \frac{0}{5} - \frac{1}{5}$

Step 3.2.1.2

Multiply $\frac{2}{1}$ by $\frac{5}{5}$ .

$y = \frac{2}{1} \cdot \frac{5}{5} + \frac{0}{5} - \frac{1}{5}$

Step 3.2.1.3

Multiply $\frac{2}{1}$ by $\frac{5}{5}$ .

$y = \frac{2 \cdot 5}{5} + \frac{0}{5} - \frac{1}{5}$

Step 3.2.2

Combine the numerators over the common denominator.

$y = \frac{2 \cdot 5 - 1}{5}$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Multiply $2$ by $5$ .

$y = \frac{10 - 1}{5}$

Step 3.2.3.2

Subtract $1$ from $10$ .

$y = \frac{9}{5}$

Step 3.3

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

$(0, \frac{9}{5}, 1)$

Step 4

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

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Step 4.1

Remove parentheses.

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

Step 4.2

Simplify $2 + \frac{1}{5} - \frac{2}{5}$ .

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Step 4.2.1

Combine the numerators over the common denominator.

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

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Subtract $2$ from $1$ .

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

Step 4.2.2.2

Move the negative in front of the fraction.

$y = 2 - \frac{1}{5}$

Step 4.2.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = 2 \cdot \frac{5}{5} - \frac{1}{5}$

Step 4.2.4

Combine $2$ and $\frac{5}{5}$ .

$y = \frac{2 \cdot 5}{5} - \frac{1}{5}$

Step 4.2.5

Combine the numerators over the common denominator.

$y = \frac{2 \cdot 5 - 1}{5}$

Step 4.2.6

Simplify the numerator.

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Step 4.2.6.1

Multiply $2$ by $5$ .

$y = \frac{10 - 1}{5}$

Step 4.2.6.2

Subtract $1$ from $10$ .

$y = \frac{9}{5}$

Step 4.3

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

$(1, \frac{9}{5}, 2)$

Step 5

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

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Step 5.1

Remove parentheses.

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

Step 5.2

Simplify $2 + \frac{2}{5} - \frac{3}{5}$ .

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Step 5.2.1

Combine the numerators over the common denominator.

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

Step 5.2.2

Simplify the expression.

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Step 5.2.2.1

Subtract $3$ from $2$ .

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

Step 5.2.2.2

Move the negative in front of the fraction.

$y = 2 - \frac{1}{5}$

Step 5.2.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = 2 \cdot \frac{5}{5} - \frac{1}{5}$

Step 5.2.4

Combine $2$ and $\frac{5}{5}$ .

$y = \frac{2 \cdot 5}{5} - \frac{1}{5}$

Step 5.2.5

Combine the numerators over the common denominator.

$y = \frac{2 \cdot 5 - 1}{5}$

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

Multiply $2$ by $5$ .

$y = \frac{10 - 1}{5}$

Step 5.2.6.2

Subtract $1$ from $10$ .

$y = \frac{9}{5}$

Step 5.3

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

$(2, \frac{9}{5}, 3)$

Step 6

These are three possible solutions to the equation.

$(0, \frac{9}{5}, 1), (1, \frac{9}{5}, 2), (2, \frac{9}{5}, 3)$