Basic Math Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Simplify terms.

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Step 1.2.1

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

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 1.3

Simplify the numerator.

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Step 1.3.1

Multiply $2$ by $5$ .

$\frac{z - 3 + 10}{5} = \frac{z + 2}{3} - 1$

Step 1.3.2

Add $- 3$ and $10$ .

$\frac{z + 7}{5} = \frac{z + 2}{3} - 1$

Step 2

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

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Step 2.1

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

$\frac{z + 7}{5} = \frac{z + 2}{3} - 1 \cdot \frac{3}{3}$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{z + 7}{5} = \frac{z + 2}{3} + \frac{- 1 \cdot 3}{3}$

Step 2.2.2

Combine the numerators over the common denominator.

$\frac{z + 7}{5} = \frac{z + 2 - 1 \cdot 3}{3}$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Multiply $- 1$ by $3$ .

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

Step 2.3.2

Subtract $3$ from $2$ .

$\frac{z + 7}{5} = \frac{z - 1}{3}$

Step 3

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

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Step 3.1

Subtract $\frac{z - 1}{3}$ from both sides of the equation.

$\frac{z + 7}{5} - \frac{z - 1}{3} = 0$

Step 3.2

To write $\frac{z + 7}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{z + 7}{5} \cdot \frac{3}{3} - \frac{z - 1}{3} = 0$

Step 3.3

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

$\frac{z + 7}{5} \cdot \frac{3}{3} - \frac{z - 1}{3} \cdot \frac{5}{5} = 0$

Step 3.4

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

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Step 3.4.1

Multiply $\frac{z + 7}{5}$ by $\frac{3}{3}$ .

$\frac{(z + 7) \cdot 3}{5 \cdot 3} - \frac{z - 1}{3} \cdot \frac{5}{5} = 0$

Step 3.4.2

Multiply $5$ by $3$ .

$\frac{(z + 7) \cdot 3}{15} - \frac{z - 1}{3} \cdot \frac{5}{5} = 0$

Step 3.4.3

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

$\frac{(z + 7) \cdot 3}{15} - \frac{(z - 1) \cdot 5}{3 \cdot 5} = 0$

Step 3.4.4

Multiply $3$ by $5$ .

$\frac{(z + 7) \cdot 3}{15} - \frac{(z - 1) \cdot 5}{15} = 0$

Step 3.5

Combine the numerators over the common denominator.

$\frac{(z + 7) \cdot 3 - (z - 1) \cdot 5}{15} = 0$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

$\frac{z \cdot 3 + 7 \cdot 3 - (z - 1) \cdot 5}{15} = 0$

Step 3.6.2

Move $3$ to the left of $z$ .

$\frac{3 \cdot z + 7 \cdot 3 - (z - 1) \cdot 5}{15} = 0$

Step 3.6.3

Multiply $7$ by $3$ .

$\frac{3 \cdot z + 21 - (z - 1) \cdot 5}{15} = 0$

Step 3.6.4

Apply the distributive property.

$\frac{3 z + 21 + (- z - - 1) \cdot 5}{15} = 0$

Step 3.6.5

Multiply $- 1$ by $- 1$ .

$\frac{3 z + 21 + (- z + 1) \cdot 5}{15} = 0$

Step 3.6.6

Apply the distributive property.

$\frac{3 z + 21 - z \cdot 5 + 1 \cdot 5}{15} = 0$

Step 3.6.7

Multiply $5$ by $- 1$ .

$\frac{3 z + 21 - 5 z + 1 \cdot 5}{15} = 0$

Step 3.6.8

Multiply $5$ by $1$ .

$\frac{3 z + 21 - 5 z + 5}{15} = 0$

Step 3.6.9

Subtract $5 z$ from $3 z$ .

$\frac{- 2 z + 21 + 5}{15} = 0$

Step 3.6.10

Add $21$ and $5$ .

$\frac{- 2 z + 26}{15} = 0$

Step 3.6.11

Factor $2$ out of $- 2 z + 26$ .

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Step 3.6.11.1

Factor $2$ out of $- 2 z$ .

$\frac{2 (- z) + 26}{15} = 0$

Step 3.6.11.2

Factor $2$ out of $26$ .

$\frac{2 (- z) + 2 (13)}{15} = 0$

Step 3.6.11.3

Factor $2$ out of $2 (- z) + 2 (13)$ .

$\frac{2 (- z + 13)}{15} = 0$

Step 3.7

Factor $- 1$ out of $- z$ .

$\frac{2 (- (z) + 13)}{15} = 0$

Step 3.8

Rewrite $13$ as $- 1 (- 13)$ .

$\frac{2 (- (z) - 1 (- 13))}{15} = 0$

Step 3.9

Factor $- 1$ out of $- (z) - 1 (- 13)$ .

$\frac{2 (- (z - 13))}{15} = 0$

Step 3.10

Rewrite $- (z - 13)$ as $- 1 (z - 13)$ .

$\frac{2 (- 1 (z - 13))}{15} = 0$

Step 3.11

Move the negative in front of the fraction.

$- \frac{2 (z - 13)}{15} = 0$

Step 4

Set the numerator equal to zero.

$2 (z - 13) = 0$

Step 5

Solve the equation for $z$ .

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Step 5.1

Divide each term in $2 (z - 13) = 0$ by $2$ and simplify.

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Step 5.1.1

Divide each term in $2 (z - 13) = 0$ by $2$ .

$\frac{2 (z - 13)}{2} = \frac{0}{2}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $2$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{2 (z - 13)}{2} = \frac{0}{2}$

Step 5.1.2.1.2

Divide $z - 13$ by $1$ .

$z - 13 = \frac{0}{2}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Divide $0$ by $2$ .

$z - 13 = 0$

Step 5.2

Add $13$ to both sides of the equation.

$z = 13$