Pre-Algebra Examples

$\frac{z}{3} - \frac{3 z}{4} = - \frac{5 z}{12}$

Step 1

Multiply both sides by $12$ .

$(\frac{z}{3} - \frac{3 z}{4}) \cdot 12 = - \frac{5 z}{12} \cdot 12$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{z}{3} - \frac{3 z}{4}) \cdot 12$ .

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Step 2.1.1.1

To write $\frac{z}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$(\frac{z}{3} \cdot \frac{4}{4} - \frac{3 z}{4}) \cdot 12 = - \frac{5 z}{12} \cdot 12$

Step 2.1.1.2

To write $- \frac{3 z}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(\frac{z}{3} \cdot \frac{4}{4} - \frac{3 z}{4} \cdot \frac{3}{3}) \cdot 12 = - \frac{5 z}{12} \cdot 12$

Step 2.1.1.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1.1.3.1

Multiply $\frac{z}{3}$ by $\frac{4}{4}$ .

$(\frac{z \cdot 4}{3 \cdot 4} - \frac{3 z}{4} \cdot \frac{3}{3}) \cdot 12 = - \frac{5 z}{12} \cdot 12$

Step 2.1.1.3.2

Multiply $3$ by $4$ .

$(\frac{z \cdot 4}{12} - \frac{3 z}{4} \cdot \frac{3}{3}) \cdot 12 = - \frac{5 z}{12} \cdot 12$

Step 2.1.1.3.3

Multiply $\frac{3 z}{4}$ by $\frac{3}{3}$ .

$(\frac{z \cdot 4}{12} - \frac{3 z \cdot 3}{4 \cdot 3}) \cdot 12 = - \frac{5 z}{12} \cdot 12$

Step 2.1.1.3.4

Multiply $4$ by $3$ .

$(\frac{z \cdot 4}{12} - \frac{3 z \cdot 3}{12}) \cdot 12 = - \frac{5 z}{12} \cdot 12$

Step 2.1.1.4

Combine the numerators over the common denominator.

$\frac{z \cdot 4 - 3 z \cdot 3}{12} \cdot 12 = - \frac{5 z}{12} \cdot 12$

Step 2.1.1.5

Simplify the numerator.

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Step 2.1.1.5.1

Factor $z$ out of $z \cdot 4 - 3 z \cdot 3$ .

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Step 2.1.1.5.1.1

Factor $z$ out of $z \cdot 4$ .

$\frac{z (4) - 3 z \cdot 3}{12} \cdot 12 = - \frac{5 z}{12} \cdot 12$

Step 2.1.1.5.1.2

Factor $z$ out of $- 3 z \cdot 3$ .

$\frac{z (4) + z (- 3 \cdot 3)}{12} \cdot 12 = - \frac{5 z}{12} \cdot 12$

Step 2.1.1.5.1.3

Factor $z$ out of $z (4) + z (- 3 \cdot 3)$ .

$\frac{z (4 - 3 \cdot 3)}{12} \cdot 12 = - \frac{5 z}{12} \cdot 12$

Step 2.1.1.5.2

Multiply $- 3$ by $3$ .

$\frac{z (4 - 9)}{12} \cdot 12 = - \frac{5 z}{12} \cdot 12$

Step 2.1.1.5.3

Subtract $9$ from $4$ .

$\frac{z \cdot - 5}{12} \cdot 12 = - \frac{5 z}{12} \cdot 12$

Step 2.1.1.6

Reduce the expression by cancelling the common factors.

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Step 2.1.1.6.1

Cancel the common factor of $12$ .

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Step 2.1.1.6.1.1

Cancel the common factor.

$\frac{z \cdot - 5}{12} \cdot 12 = - \frac{5 z}{12} \cdot 12$

Step 2.1.1.6.1.2

Rewrite the expression.

$z \cdot - 5 = - \frac{5 z}{12} \cdot 12$

Step 2.1.1.6.2

Move $- 5$ to the left of $z$ .

$- 5 z = - \frac{5 z}{12} \cdot 12$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $12$ .

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Step 2.2.1.1

Move the leading negative in $- \frac{5 z}{12}$ into the numerator.

$- 5 z = \frac{- 5 z}{12} \cdot 12$

Step 2.2.1.2

Cancel the common factor.

$- 5 z = \frac{- 5 z}{12} \cdot 12$

Step 2.2.1.3

Rewrite the expression.

$- 5 z = - 5 z$

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

Add $5 z$ to both sides of the equation.

$- 5 z + 5 z = 0$

Step 3.1.2

Add $- 5 z$ and $5 z$ .

$0 = 0$

Step 3.2

Since $0 = 0$ , the equation will always be true.

Always true