Algebra Examples

$\frac{1}{3} x y + \frac{5}{4} x y^{2} - \frac{3}{4} x y + \frac{1}{2} x y^{2} - 3 x^{2} y + \frac{5}{6} x^{2} y$

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 $\frac{1}{3}$ .

$\frac{x}{3} y + \frac{5}{4} \cdot (x y^{2}) - \frac{3}{4} \cdot (x y) + \frac{1}{2} \cdot (x y^{2}) - 3 x^{2} y + \frac{5}{6} \cdot (x^{2} y)$

Step 1.1.2

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

$\frac{x y}{3} + \frac{5}{4} \cdot (x y^{2}) - \frac{3}{4} \cdot (x y) + \frac{1}{2} \cdot (x y^{2}) - 3 x^{2} y + \frac{5}{6} \cdot (x^{2} y)$

Step 1.2

Multiply $\frac{5}{4} (x y^{2})$ .

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Step 1.2.1

Combine $x$ and $\frac{5}{4}$ .

$\frac{x y}{3} + \frac{x \cdot 5}{4} y^{2} - \frac{3}{4} \cdot (x y) + \frac{1}{2} \cdot (x y^{2}) - 3 x^{2} y + \frac{5}{6} \cdot (x^{2} y)$

Step 1.2.2

Combine $\frac{x \cdot 5}{4}$ and $y^{2}$ .

$\frac{x y}{3} + \frac{x \cdot 5 y^{2}}{4} - \frac{3}{4} \cdot (x y) + \frac{1}{2} \cdot (x y^{2}) - 3 x^{2} y + \frac{5}{6} \cdot (x^{2} y)$

Step 1.3

Move $5$ to the left of $x$ .

$\frac{x y}{3} + \frac{5 \cdot x y^{2}}{4} - \frac{3}{4} \cdot (x y) + \frac{1}{2} \cdot (x y^{2}) - 3 x^{2} y + \frac{5}{6} \cdot (x^{2} y)$

Step 1.4

Multiply $- \frac{3}{4} (x y)$ .

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Step 1.4.1

Combine $x$ and $\frac{3}{4}$ .

$\frac{x y}{3} + \frac{5 x y^{2}}{4} - \frac{x \cdot 3}{4} y + \frac{1}{2} \cdot (x y^{2}) - 3 x^{2} y + \frac{5}{6} \cdot (x^{2} y)$

Step 1.4.2

Combine $y$ and $\frac{x \cdot 3}{4}$ .

$\frac{x y}{3} + \frac{5 x y^{2}}{4} - \frac{y (x \cdot 3)}{4} + \frac{1}{2} \cdot (x y^{2}) - 3 x^{2} y + \frac{5}{6} \cdot (x^{2} y)$

Step 1.5

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

$\frac{x y}{3} + \frac{5 x y^{2}}{4} - \frac{3 \cdot (y x)}{4} + \frac{1}{2} \cdot (x y^{2}) - 3 x^{2} y + \frac{5}{6} \cdot (x^{2} y)$

Step 1.6

Multiply $\frac{1}{2} (x y^{2})$ .

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Step 1.6.1

Combine $x$ and $\frac{1}{2}$ .

$\frac{x y}{3} + \frac{5 x y^{2}}{4} - \frac{3 y x}{4} + \frac{x}{2} y^{2} - 3 x^{2} y + \frac{5}{6} \cdot (x^{2} y)$

Step 1.6.2

Combine $\frac{x}{2}$ and $y^{2}$ .

$\frac{x y}{3} + \frac{5 x y^{2}}{4} - \frac{3 y x}{4} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5}{6} \cdot (x^{2} y)$

Step 1.7

Multiply $\frac{5}{6} (x^{2} y)$ .

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Step 1.7.1

Combine $x^{2}$ and $\frac{5}{6}$ .

$\frac{x y}{3} + \frac{5 x y^{2}}{4} - \frac{3 y x}{4} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{x^{2} \cdot 5}{6} y$

Step 1.7.2

Combine $\frac{x^{2} \cdot 5}{6}$ and $y$ .

$\frac{x y}{3} + \frac{5 x y^{2}}{4} - \frac{3 y x}{4} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{x^{2} \cdot 5 y}{6}$

Step 1.8

Move $5$ to the left of $x^{2}$ .

$\frac{x y}{3} + \frac{5 x y^{2}}{4} - \frac{3 y x}{4} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 2

Subtract $\frac{3 y x}{4}$ from $\frac{x y}{3}$ .

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Step 2.1

Move $y$ .

$\frac{5 x y^{2}}{4} + \frac{x y}{3} - \frac{3 x y}{4} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 2.2

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

$\frac{5 x y^{2}}{4} + \frac{x y}{3} \cdot \frac{4}{4} - \frac{3 x y}{4} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 2.3

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

$\frac{5 x y^{2}}{4} + \frac{x y}{3} \cdot \frac{4}{4} - \frac{3 x y}{4} \cdot \frac{3}{3} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 2.4

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

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Step 2.4.1

Multiply $\frac{x y}{3}$ by $\frac{4}{4}$ .

$\frac{5 x y^{2}}{4} + \frac{x y \cdot 4}{3 \cdot 4} - \frac{3 x y}{4} \cdot \frac{3}{3} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 2.4.2

Multiply $3$ by $4$ .

$\frac{5 x y^{2}}{4} + \frac{x y \cdot 4}{12} - \frac{3 x y}{4} \cdot \frac{3}{3} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 2.4.3

Multiply $\frac{3 x y}{4}$ by $\frac{3}{3}$ .

$\frac{5 x y^{2}}{4} + \frac{x y \cdot 4}{12} - \frac{3 x y \cdot 3}{4 \cdot 3} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 2.4.4

Multiply $4$ by $3$ .

$\frac{5 x y^{2}}{4} + \frac{x y \cdot 4}{12} - \frac{3 x y \cdot 3}{12} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{5 x y^{2}}{4} + \frac{x y \cdot 4 - 3 x y \cdot 3}{12} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 3

Simplify each term.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Factor $x y$ out of $x y \cdot 4 - 3 x y \cdot 3$ .

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Step 3.1.1.1

Factor $x y$ out of $x y \cdot 4$ .

$\frac{5 x y^{2}}{4} + \frac{x y (4) - 3 x y \cdot 3}{12} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 3.1.1.2

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

$\frac{5 x y^{2}}{4} + \frac{x y (4) + x y (- 3 \cdot 3)}{12} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 3.1.1.3

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

$\frac{5 x y^{2}}{4} + \frac{x y (4 - 3 \cdot 3)}{12} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 3.1.2

Multiply $- 3$ by $3$ .

$\frac{5 x y^{2}}{4} + \frac{x y (4 - 9)}{12} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 3.1.3

Subtract $9$ from $4$ .

$\frac{5 x y^{2}}{4} + \frac{x y \cdot - 5}{12} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 3.2

Move $- 5$ to the left of $x y$ .

$\frac{5 x y^{2}}{4} + \frac{- 5 \cdot (x y)}{12} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 3.3

Move the negative in front of the fraction.

$\frac{5 x y^{2}}{4} - \frac{5 x y}{12} + \frac{x y^{2}}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 4

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

$- \frac{5 x y}{12} + \frac{5 x y^{2}}{4} + \frac{x y^{2}}{2} \cdot \frac{2}{2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 5

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{x y^{2}}{2}$ by $\frac{2}{2}$ .

$- \frac{5 x y}{12} + \frac{5 x y^{2}}{4} + \frac{x y^{2} \cdot 2}{2 \cdot 2} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 5.2

Multiply $2$ by $2$ .

$- \frac{5 x y}{12} + \frac{5 x y^{2}}{4} + \frac{x y^{2} \cdot 2}{4} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 6

Combine the numerators over the common denominator.

$- \frac{5 x y}{12} + \frac{5 x y^{2} + x y^{2} \cdot 2}{4} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 7

Add $5 x y^{2}$ and $x y^{2} \cdot 2$ .

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Step 7.1

Reorder $x$ and $2$ .

$- \frac{5 x y}{12} + \frac{5 x y^{2} + 2 \cdot x y^{2}}{4} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 7.2

Add $5 x y^{2}$ and $2 \cdot x y^{2}$ .

$- \frac{5 x y}{12} + \frac{7 x y^{2}}{4} - 3 x^{2} y + \frac{5 x^{2} y}{6}$

Step 8

To write $- 3 x^{2} y$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{5 x y}{12} + \frac{7 x y^{2}}{4} - 3 x^{2} y \cdot \frac{6}{6} + \frac{5 x^{2} y}{6}$

Step 9

Simplify terms.

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Step 9.1

Combine $- 3 x^{2} y$ and $\frac{6}{6}$ .

$- \frac{5 x y}{12} + \frac{7 x y^{2}}{4} + \frac{- 3 x^{2} y \cdot 6}{6} + \frac{5 x^{2} y}{6}$

Step 9.2

Combine the numerators over the common denominator.

$- \frac{5 x y}{12} + \frac{7 x y^{2}}{4} + \frac{- 3 x^{2} y \cdot 6 + 5 x^{2} y}{6}$

Step 10

Simplify each term.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Factor $x^{2} y$ out of $- 3 x^{2} y \cdot 6 + 5 x^{2} y$ .

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Step 10.1.1.1

Factor $x^{2} y$ out of $- 3 x^{2} y \cdot 6$ .

$- \frac{5 x y}{12} + \frac{7 x y^{2}}{4} + \frac{x^{2} y (- 3 \cdot 6) + 5 x^{2} y}{6}$

Step 10.1.1.2

Factor $x^{2} y$ out of $5 x^{2} y$ .

$- \frac{5 x y}{12} + \frac{7 x y^{2}}{4} + \frac{x^{2} y (- 3 \cdot 6) + x^{2} y (5)}{6}$

Step 10.1.1.3

Factor $x^{2} y$ out of $x^{2} y (- 3 \cdot 6) + x^{2} y (5)$ .

$- \frac{5 x y}{12} + \frac{7 x y^{2}}{4} + \frac{x^{2} y (- 3 \cdot 6 + 5)}{6}$

Step 10.1.2

Multiply $- 3$ by $6$ .

$- \frac{5 x y}{12} + \frac{7 x y^{2}}{4} + \frac{x^{2} y (- 18 + 5)}{6}$

Step 10.1.3

Add $- 18$ and $5$ .

$- \frac{5 x y}{12} + \frac{7 x y^{2}}{4} + \frac{x^{2} y \cdot - 13}{6}$

Step 10.2

Move $- 13$ to the left of $x^{2} y$ .

$- \frac{5 x y}{12} + \frac{7 x y^{2}}{4} + \frac{- 13 \cdot (x^{2} y)}{6}$

Step 10.3

Move the negative in front of the fraction.

$- \frac{5 x y}{12} + \frac{7 x y^{2}}{4} - \frac{13 x^{2} y}{6}$