Precalculus Examples

$\frac{z}{5} = \frac{3}{10} z + 7$

Step 1

Multiply both sides by $5$ .

$\frac{z}{5} \cdot 5 = (\frac{3}{10} z + 7) \cdot 5$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $5$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{z}{5} \cdot 5 = (\frac{3}{10} z + 7) \cdot 5$

Step 2.1.1.2

Rewrite the expression.

$z = (\frac{3}{10} z + 7) \cdot 5$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\frac{3}{10} z + 7) \cdot 5$ .

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Step 2.2.1.1

Combine $\frac{3}{10}$ and $z$ .

$z = (\frac{3 z}{10} + 7) \cdot 5$

Step 2.2.1.2

Apply the distributive property.

$z = \frac{3 z}{10} \cdot 5 + 7 \cdot 5$

Step 2.2.1.3

Cancel the common factor of $5$ .

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Step 2.2.1.3.1

Factor $5$ out of $10$ .

$z = \frac{3 z}{5 (2)} \cdot 5 + 7 \cdot 5$

Step 2.2.1.3.2

Cancel the common factor.

$z = \frac{3 z}{5 \cdot 2} \cdot 5 + 7 \cdot 5$

Step 2.2.1.3.3

Rewrite the expression.

$z = \frac{3 z}{2} + 7 \cdot 5$

Step 2.2.1.4

Multiply $7$ by $5$ .

$z = \frac{3 z}{2} + 35$

Step 3

Solve for $z$ .

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Step 3.1

Move all terms containing $z$ to the left side of the equation.

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Step 3.1.1

Subtract $\frac{3 z}{2}$ from both sides of the equation.

$z - \frac{3 z}{2} = 35$

Step 3.1.2

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

$z \cdot \frac{2}{2} - \frac{3 z}{2} = 35$

Step 3.1.3

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

$\frac{z \cdot 2}{2} - \frac{3 z}{2} = 35$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{z \cdot 2 - 3 z}{2} = 35$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Move $2$ to the left of $z$ .

$\frac{2 \cdot z - 3 z}{2} = 35$

Step 3.1.5.2

Subtract $3 z$ from $2 z$ .

$\frac{- z}{2} = 35$

Step 3.1.6

Move the negative in front of the fraction.

$- \frac{z}{2} = 35$

Step 3.2

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{z}{2}) = - 2 \cdot 35$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

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

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Step 3.3.1.1.1

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.1

Move the leading negative in $- \frac{z}{2}$ into the numerator.

$- 2 \frac{- z}{2} = - 2 \cdot 35$

Step 3.3.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- z}{2} = - 2 \cdot 35$

Step 3.3.1.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- z}{2} = - 2 \cdot 35$

Step 3.3.1.1.1.4

Rewrite the expression.

$- - z = - 2 \cdot 35$

Step 3.3.1.1.2

Multiply.

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Step 3.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 z = - 2 \cdot 35$

Step 3.3.1.1.2.2

Multiply $z$ by $1$ .

$z = - 2 \cdot 35$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $- 2$ by $35$ .

$z = - 70$