Basic Math Examples

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

Step 1

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

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

Step 2

Write each expression with a common denominator of $z \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1

Multiply $\frac{5 y}{z}$ by $\frac{2}{2}$ .

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

Step 2.2

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

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $y$ out of $5 y \cdot 2 - 3 y$ .

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Step 4.1.1.1

Factor $y$ out of $5 y \cdot 2$ .

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

Step 4.1.1.2

Factor $y$ out of $- 3 y$ .

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

Step 4.1.1.3

Factor $y$ out of $y (5 \cdot 2) + y \cdot - 3$ .

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

Step 4.1.2

Multiply $5$ by $2$ .

$\frac{y (10 - 3)}{2 z} + \frac{y}{3 z}$

Step 4.1.3

Subtract $3$ from $10$ .

$\frac{y \cdot 7}{2 z} + \frac{y}{3 z}$

Step 4.2

Move $7$ to the left of $y$ .

$\frac{7 y}{2 z} + \frac{y}{3 z}$

Step 5

To write $\frac{7 y}{2 z}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{7 y}{2 z} \cdot \frac{3}{3} + \frac{y}{3 z}$

Step 6

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

$\frac{7 y}{2 z} \cdot \frac{3}{3} + \frac{y}{3 z} \cdot \frac{2}{2}$

Step 7

Write each expression with a common denominator of $6 z$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{7 y}{2 z}$ by $\frac{3}{3}$ .

$\frac{7 y \cdot 3}{2 z \cdot 3} + \frac{y}{3 z} \cdot \frac{2}{2}$

Step 7.2

Multiply $3$ by $2$ .

$\frac{7 y \cdot 3}{6 z} + \frac{y}{3 z} \cdot \frac{2}{2}$

Step 7.3

Multiply $\frac{y}{3 z}$ by $\frac{2}{2}$ .

$\frac{7 y \cdot 3}{6 z} + \frac{y \cdot 2}{3 z \cdot 2}$

Step 7.4

Multiply $2$ by $3$ .

$\frac{7 y \cdot 3}{6 z} + \frac{y \cdot 2}{6 z}$

Step 8

Combine the numerators over the common denominator.

$\frac{7 y \cdot 3 + y \cdot 2}{6 z}$

Step 9

Simplify the numerator.

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Step 9.1

Factor $y$ out of $7 y \cdot 3 + y \cdot 2$ .

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Step 9.1.1

Factor $y$ out of $7 y \cdot 3$ .

$\frac{y (7 \cdot 3) + y \cdot 2}{6 z}$

Step 9.1.2

Factor $y$ out of $y \cdot 2$ .

$\frac{y (7 \cdot 3) + y (2)}{6 z}$

Step 9.1.3

Factor $y$ out of $y (7 \cdot 3) + y (2)$ .

$\frac{y (7 \cdot 3 + 2)}{6 z}$

Step 9.2

Multiply $7$ by $3$ .

$\frac{y (21 + 2)}{6 z}$

Step 9.3

Add $21$ and $2$ .

$\frac{y \cdot 23}{6 z}$

Step 10

Move $23$ to the left of $y$ .

$\frac{23 y}{6 z}$