Algebra Examples

$\frac{2 z - 1}{3} - \frac{3 z + 1}{5} = 1$

Step 1

Simplify $\frac{2 z - 1}{3} - \frac{3 z + 1}{5}$ .

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Step 1.1

To write $\frac{2 z - 1}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2 z - 1}{3} \cdot \frac{5}{5} - \frac{3 z + 1}{5} = 1$

Step 1.2

To write $- \frac{3 z + 1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 z - 1}{3} \cdot \frac{5}{5} - \frac{3 z + 1}{5} \cdot \frac{3}{3} = 1$

Step 1.3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{2 z - 1}{3}$ by $\frac{5}{5}$ .

$\frac{(2 z - 1) \cdot 5}{3 \cdot 5} - \frac{3 z + 1}{5} \cdot \frac{3}{3} = 1$

Step 1.3.2

Multiply $3$ by $5$ .

$\frac{(2 z - 1) \cdot 5}{15} - \frac{3 z + 1}{5} \cdot \frac{3}{3} = 1$

Step 1.3.3

Multiply $\frac{3 z + 1}{5}$ by $\frac{3}{3}$ .

$\frac{(2 z - 1) \cdot 5}{15} - \frac{(3 z + 1) \cdot 3}{5 \cdot 3} = 1$

Step 1.3.4

Multiply $5$ by $3$ .

$\frac{(2 z - 1) \cdot 5}{15} - \frac{(3 z + 1) \cdot 3}{15} = 1$

Step 1.4

Combine the numerators over the common denominator.

$\frac{(2 z - 1) \cdot 5 - (3 z + 1) \cdot 3}{15} = 1$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{2 z \cdot 5 - 1 \cdot 5 - (3 z + 1) \cdot 3}{15} = 1$

Step 1.5.2

Multiply $5$ by $2$ .

$\frac{10 z - 1 \cdot 5 - (3 z + 1) \cdot 3}{15} = 1$

Step 1.5.3

Multiply $- 1$ by $5$ .

$\frac{10 z - 5 - (3 z + 1) \cdot 3}{15} = 1$

Step 1.5.4

Apply the distributive property.

$\frac{10 z - 5 + (- (3 z) - 1 \cdot 1) \cdot 3}{15} = 1$

Step 1.5.5

Multiply $3$ by $- 1$ .

$\frac{10 z - 5 + (- 3 z - 1 \cdot 1) \cdot 3}{15} = 1$

Step 1.5.6

Multiply $- 1$ by $1$ .

$\frac{10 z - 5 + (- 3 z - 1) \cdot 3}{15} = 1$

Step 1.5.7

Apply the distributive property.

$\frac{10 z - 5 - 3 z \cdot 3 - 1 \cdot 3}{15} = 1$

Step 1.5.8

Multiply $3$ by $- 3$ .

$\frac{10 z - 5 - 9 z - 1 \cdot 3}{15} = 1$

Step 1.5.9

Multiply $- 1$ by $3$ .

$\frac{10 z - 5 - 9 z - 3}{15} = 1$

Step 1.5.10

Subtract $9 z$ from $10 z$ .

$\frac{z - 5 - 3}{15} = 1$

Step 1.5.11

Subtract $3$ from $- 5$ .

$\frac{z - 8}{15} = 1$

Step 2

Multiply both sides of the equation by $15$ .

$15 \frac{z - 8}{15} = 15 \cdot 1$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $15$ .

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Step 3.1.1.1

Cancel the common factor.

$15 \frac{z - 8}{15} = 15 \cdot 1$

Step 3.1.1.2

Rewrite the expression.

$z - 8 = 15 \cdot 1$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $15$ by $1$ .

$z - 8 = 15$

Step 4

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

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Step 4.1

Add $8$ to both sides of the equation.

$z = 15 + 8$

Step 4.2

Add $15$ and $8$ .

$z = 23$