Algebra Examples

$\frac{1}{3} x y + \frac{2}{5} x y - 2 x z + \frac{3}{2} x y + \frac{4}{9} x z$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{1}{3} (x y)$ .

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Step 1.1.1

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

$\frac{x}{3} y + \frac{2}{5} \cdot (x y) - 2 x z + \frac{3}{2} \cdot (x y) + \frac{4}{9} \cdot (x z)$

Step 1.1.2

Combine $\frac{x}{3}$ and $y$ .

$\frac{x y}{3} + \frac{2}{5} \cdot (x y) - 2 x z + \frac{3}{2} \cdot (x y) + \frac{4}{9} \cdot (x z)$

Step 1.2

Multiply $\frac{2}{5} (x y)$ .

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Step 1.2.1

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

$\frac{x y}{3} + \frac{x \cdot 2}{5} y - 2 x z + \frac{3}{2} \cdot (x y) + \frac{4}{9} \cdot (x z)$

Step 1.2.2

Combine $\frac{x \cdot 2}{5}$ and $y$ .

$\frac{x y}{3} + \frac{x \cdot 2 y}{5} - 2 x z + \frac{3}{2} \cdot (x y) + \frac{4}{9} \cdot (x z)$

Step 1.3

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

$\frac{x y}{3} + \frac{2 \cdot x y}{5} - 2 x z + \frac{3}{2} \cdot (x y) + \frac{4}{9} \cdot (x z)$

Step 1.4

Multiply $\frac{3}{2} (x y)$ .

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Step 1.4.1

Combine $x$ and $\frac{3}{2}$ .

$\frac{x y}{3} + \frac{2 x y}{5} - 2 x z + \frac{x \cdot 3}{2} y + \frac{4}{9} \cdot (x z)$

Step 1.4.2

Combine $\frac{x \cdot 3}{2}$ and $y$ .

$\frac{x y}{3} + \frac{2 x y}{5} - 2 x z + \frac{x \cdot 3 y}{2} + \frac{4}{9} \cdot (x z)$

Step 1.5

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

$\frac{x y}{3} + \frac{2 x y}{5} - 2 x z + \frac{3 \cdot x y}{2} + \frac{4}{9} \cdot (x z)$

Step 1.6

Multiply $\frac{4}{9} (x z)$ .

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Step 1.6.1

Combine $x$ and $\frac{4}{9}$ .

$\frac{x y}{3} + \frac{2 x y}{5} - 2 x z + \frac{3 x y}{2} + \frac{x \cdot 4}{9} z$

Step 1.6.2

Combine $\frac{x \cdot 4}{9}$ and $z$ .

$\frac{x y}{3} + \frac{2 x y}{5} - 2 x z + \frac{3 x y}{2} + \frac{x \cdot 4 z}{9}$

Step 1.7

Move $4$ to the left of $x$ .

$\frac{x y}{3} + \frac{2 x y}{5} - 2 x z + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 2

To write $\frac{x y}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x y}{3} \cdot \frac{5}{5} + \frac{2 x y}{5} - 2 x z + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 3

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

$\frac{x y}{3} \cdot \frac{5}{5} + \frac{2 x y}{5} \cdot \frac{3}{3} - 2 x z + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 4

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

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Step 4.1

Multiply $\frac{x y}{3}$ by $\frac{5}{5}$ .

$\frac{x y \cdot 5}{3 \cdot 5} + \frac{2 x y}{5} \cdot \frac{3}{3} - 2 x z + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 4.2

Multiply $3$ by $5$ .

$\frac{x y \cdot 5}{15} + \frac{2 x y}{5} \cdot \frac{3}{3} - 2 x z + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 4.3

Multiply $\frac{2 x y}{5}$ by $\frac{3}{3}$ .

$\frac{x y \cdot 5}{15} + \frac{2 x y \cdot 3}{5 \cdot 3} - 2 x z + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 4.4

Multiply $5$ by $3$ .

$\frac{x y \cdot 5}{15} + \frac{2 x y \cdot 3}{15} - 2 x z + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 5

Combine the numerators over the common denominator.

$\frac{x y \cdot 5 + 2 x y \cdot 3}{15} - 2 x z + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 6

Simplify each term.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Factor $x y$ out of $x y \cdot 5 + 2 x y \cdot 3$ .

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Step 6.1.1.1

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

$\frac{x y (5) + 2 x y \cdot 3}{15} - 2 x z + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 6.1.1.2

Factor $x y$ out of $2 x y \cdot 3$ .

$\frac{x y (5) + x y (2 \cdot 3)}{15} - 2 x z + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 6.1.1.3

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

$\frac{x y (5 + 2 \cdot 3)}{15} - 2 x z + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 6.1.2

Multiply $2$ by $3$ .

$\frac{x y (5 + 6)}{15} - 2 x z + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 6.1.3

Add $5$ and $6$ .

$\frac{x y \cdot 11}{15} - 2 x z + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 6.2

Move $11$ to the left of $x y$ .

$\frac{11 x y}{15} - 2 x z + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 7

To write $\frac{11 x y}{15}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 2 x z + \frac{11 x y}{15} \cdot \frac{2}{2} + \frac{3 x y}{2} + \frac{4 x z}{9}$

Step 8

To write $\frac{3 x y}{2}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$- 2 x z + \frac{11 x y}{15} \cdot \frac{2}{2} + \frac{3 x y}{2} \cdot \frac{15}{15} + \frac{4 x z}{9}$

Step 9

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

Multiply $\frac{11 x y}{15}$ by $\frac{2}{2}$ .

$- 2 x z + \frac{11 x y \cdot 2}{15 \cdot 2} + \frac{3 x y}{2} \cdot \frac{15}{15} + \frac{4 x z}{9}$

Step 9.2

Multiply $15$ by $2$ .

$- 2 x z + \frac{11 x y \cdot 2}{30} + \frac{3 x y}{2} \cdot \frac{15}{15} + \frac{4 x z}{9}$

Step 9.3

Multiply $\frac{3 x y}{2}$ by $\frac{15}{15}$ .

$- 2 x z + \frac{11 x y \cdot 2}{30} + \frac{3 x y \cdot 15}{2 \cdot 15} + \frac{4 x z}{9}$

Step 9.4

Multiply $2$ by $15$ .

$- 2 x z + \frac{11 x y \cdot 2}{30} + \frac{3 x y \cdot 15}{30} + \frac{4 x z}{9}$

Step 10

Combine the numerators over the common denominator.

$- 2 x z + \frac{11 x y \cdot 2 + 3 x y \cdot 15}{30} + \frac{4 x z}{9}$

Step 11

Simplify each term.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Factor $x y$ out of $11 x y \cdot 2 + 3 x y \cdot 15$ .

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Step 11.1.1.1

Factor $x y$ out of $11 x y \cdot 2$ .

$- 2 x z + \frac{x y (11 \cdot 2) + 3 x y \cdot 15}{30} + \frac{4 x z}{9}$

Step 11.1.1.2

Factor $x y$ out of $3 x y \cdot 15$ .

$- 2 x z + \frac{x y (11 \cdot 2) + x y (3 \cdot 15)}{30} + \frac{4 x z}{9}$

Step 11.1.1.3

Factor $x y$ out of $x y (11 \cdot 2) + x y (3 \cdot 15)$ .

$- 2 x z + \frac{x y (11 \cdot 2 + 3 \cdot 15)}{30} + \frac{4 x z}{9}$

Step 11.1.2

Combine exponents.

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Step 11.1.2.1

Multiply $11$ by $2$ .

$- 2 x z + \frac{x y (22 + 3 \cdot 15)}{30} + \frac{4 x z}{9}$

Step 11.1.2.2

Multiply $3$ by $15$ .

$- 2 x z + \frac{x y (22 + 45)}{30} + \frac{4 x z}{9}$

Step 11.1.3

Add $22$ and $45$ .

$- 2 x z + \frac{x y \cdot 67}{30} + \frac{4 x z}{9}$

Step 11.2

Move $67$ to the left of $x y$ .

$- 2 x z + \frac{67 x y}{30} + \frac{4 x z}{9}$

Step 12

To write $- 2 x z$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{67 x y}{30} - 2 x z \cdot \frac{9}{9} + \frac{4 x z}{9}$

Step 13

Simplify terms.

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Step 13.1

Combine $- 2 x z$ and $\frac{9}{9}$ .

$\frac{67 x y}{30} + \frac{- 2 x z \cdot 9}{9} + \frac{4 x z}{9}$

Step 13.2

Combine the numerators over the common denominator.

$\frac{67 x y}{30} + \frac{- 2 x z \cdot 9 + 4 x z}{9}$

Step 14

Simplify each term.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

Factor $2 x z$ out of $- 2 x z \cdot 9 + 4 x z$ .

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Step 14.1.1.1

Factor $2 x z$ out of $- 2 x z \cdot 9$ .

$\frac{67 x y}{30} + \frac{2 x z (- 1 \cdot 9) + 4 x z}{9}$

Step 14.1.1.2

Factor $2 x z$ out of $4 x z$ .

$\frac{67 x y}{30} + \frac{2 x z (- 1 \cdot 9) + 2 x z (2)}{9}$

Step 14.1.1.3

Factor $2 x z$ out of $2 x z (- 1 \cdot 9) + 2 x z (2)$ .

$\frac{67 x y}{30} + \frac{2 x z (- 1 \cdot 9 + 2)}{9}$

Step 14.1.2

Multiply $- 1$ by $9$ .

$\frac{67 x y}{30} + \frac{2 x z (- 9 + 2)}{9}$

Step 14.1.3

Add $- 9$ and $2$ .

$\frac{67 x y}{30} + \frac{2 x z \cdot - 7}{9}$

Step 14.1.4

Multiply $- 7$ by $2$ .

$\frac{67 x y}{30} + \frac{- 14 x z}{9}$

Step 14.2

Move the negative in front of the fraction.

$\frac{67 x y}{30} - \frac{14 x z}{9}$