Pre-Algebra Examples

$\frac{\frac{1}{3} x^{2} + \frac{7}{10} \cdot (x y) - \frac{1}{3} y^{2}}{x - \frac{2}{5} y}$

Step 1

Simplify the numerator.

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Step 1.1

Combine $\frac{1}{3}$ and $x^{2}$ .

$\frac{\frac{x^{2}}{3} + \frac{7}{10} \cdot x y - \frac{1}{3} y^{2}}{x - \frac{2}{5} y}$

Step 1.2

Combine $\frac{7}{10}$ and $x$ .

$\frac{\frac{x^{2}}{3} + \frac{7 x}{10} y - \frac{1}{3} y^{2}}{x - \frac{2}{5} y}$

Step 1.3

Combine $\frac{7 x}{10}$ and $y$ .

$\frac{\frac{x^{2}}{3} + \frac{7 x y}{10} - \frac{1}{3} y^{2}}{x - \frac{2}{5} y}$

Step 1.4

Combine $y^{2}$ and $\frac{1}{3}$ .

$\frac{\frac{x^{2}}{3} + \frac{7 x y}{10} - \frac{y^{2}}{3}}{x - \frac{2}{5} y}$

Step 1.5

To write $\frac{x^{2}}{3}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{\frac{x^{2}}{3} \cdot \frac{10}{10} + \frac{7 x y}{10} - \frac{y^{2}}{3}}{x - \frac{2}{5} y}$

Step 1.6

To write $\frac{7 x y}{10}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{x^{2}}{3} \cdot \frac{10}{10} + \frac{7 x y}{10} \cdot \frac{3}{3} - \frac{y^{2}}{3}}{x - \frac{2}{5} y}$

Step 1.7

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

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Step 1.7.1

Multiply $\frac{x^{2}}{3}$ by $\frac{10}{10}$ .

$\frac{\frac{x^{2} \cdot 10}{3 \cdot 10} + \frac{7 x y}{10} \cdot \frac{3}{3} - \frac{y^{2}}{3}}{x - \frac{2}{5} y}$

Step 1.7.2

Multiply $3$ by $10$ .

$\frac{\frac{x^{2} \cdot 10}{30} + \frac{7 x y}{10} \cdot \frac{3}{3} - \frac{y^{2}}{3}}{x - \frac{2}{5} y}$

Step 1.7.3

Multiply $\frac{7 x y}{10}$ by $\frac{3}{3}$ .

$\frac{\frac{x^{2} \cdot 10}{30} + \frac{7 x y \cdot 3}{10 \cdot 3} - \frac{y^{2}}{3}}{x - \frac{2}{5} y}$

Step 1.7.4

Multiply $10$ by $3$ .

$\frac{\frac{x^{2} \cdot 10}{30} + \frac{7 x y \cdot 3}{30} - \frac{y^{2}}{3}}{x - \frac{2}{5} y}$

Step 1.8

Combine the numerators over the common denominator.

$\frac{\frac{x^{2} \cdot 10 + 7 x y \cdot 3}{30} - \frac{y^{2}}{3}}{x - \frac{2}{5} y}$

Step 1.9

Simplify the numerator.

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Step 1.9.1

Factor $x$ out of $x^{2} \cdot 10 + 7 x y \cdot 3$ .

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Step 1.9.1.1

Factor $x$ out of $x^{2} \cdot 10$ .

$\frac{\frac{x (x \cdot 10) + 7 x y \cdot 3}{30} - \frac{y^{2}}{3}}{x - \frac{2}{5} y}$

Step 1.9.1.2

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

$\frac{\frac{x (x \cdot 10) + x (7 y \cdot 3)}{30} - \frac{y^{2}}{3}}{x - \frac{2}{5} y}$

Step 1.9.1.3

Factor $x$ out of $x (x \cdot 10) + x (7 y \cdot 3)$ .

$\frac{\frac{x (x \cdot 10 + 7 y \cdot 3)}{30} - \frac{y^{2}}{3}}{x - \frac{2}{5} y}$

Step 1.9.2

Move $10$ to the left of $x$ .

$\frac{\frac{x (10 \cdot x + 7 y \cdot 3)}{30} - \frac{y^{2}}{3}}{x - \frac{2}{5} y}$

Step 1.9.3

Multiply $3$ by $7$ .

$\frac{\frac{x (10 x + 21 y)}{30} - \frac{y^{2}}{3}}{x - \frac{2}{5} y}$

Step 1.10

To write $- \frac{y^{2}}{3}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{\frac{x (10 x + 21 y)}{30} - \frac{y^{2}}{3} \cdot \frac{10}{10}}{x - \frac{2}{5} y}$

Step 1.11

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

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Step 1.11.1

Multiply $\frac{y^{2}}{3}$ by $\frac{10}{10}$ .

$\frac{\frac{x (10 x + 21 y)}{30} - \frac{y^{2} \cdot 10}{3 \cdot 10}}{x - \frac{2}{5} y}$

Step 1.11.2

Multiply $3$ by $10$ .

$\frac{\frac{x (10 x + 21 y)}{30} - \frac{y^{2} \cdot 10}{30}}{x - \frac{2}{5} y}$

Step 1.12

Combine the numerators over the common denominator.

$\frac{\frac{x (10 x + 21 y) - y^{2} \cdot 10}{30}}{x - \frac{2}{5} y}$

Step 1.13

Simplify the numerator.

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Step 1.13.1

Apply the distributive property.

$\frac{\frac{x (10 x) + x (21 y) - y^{2} \cdot 10}{30}}{x - \frac{2}{5} y}$

Step 1.13.2

Rewrite using the commutative property of multiplication.

$\frac{\frac{10 x \cdot x + x (21 y) - y^{2} \cdot 10}{30}}{x - \frac{2}{5} y}$

Step 1.13.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{10 x \cdot x + 21 x y - y^{2} \cdot 10}{30}}{x - \frac{2}{5} y}$

Step 1.13.4

Multiply $x$ by $x$ by adding the exponents.

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Step 1.13.4.1

Move $x$ .

$\frac{\frac{10 (x \cdot x) + 21 x y - y^{2} \cdot 10}{30}}{x - \frac{2}{5} y}$

Step 1.13.4.2

Multiply $x$ by $x$ .

$\frac{\frac{10 x^{2} + 21 x y - y^{2} \cdot 10}{30}}{x - \frac{2}{5} y}$

Step 1.13.5

Multiply $10$ by $- 1$ .

$\frac{\frac{10 x^{2} + 21 x y - 10 y^{2}}{30}}{x - \frac{2}{5} y}$

Step 1.13.6

Factor by grouping.

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Step 1.13.6.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 10 \cdot - 10 = - 100$ and whose sum is $b = 21$ .

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Step 1.13.6.1.1

Reorder terms.

$\frac{\frac{10 x^{2} - 10 y^{2} + 21 x y}{30}}{x - \frac{2}{5} y}$

Step 1.13.6.1.2

Reorder $- 10 y^{2}$ and $21 x y$ .

$\frac{\frac{10 x^{2} + 21 x y - 10 y^{2}}{30}}{x - \frac{2}{5} y}$

Step 1.13.6.1.3

Factor $21$ out of $21 x y$ .

$\frac{\frac{10 x^{2} + 21 (x y) - 10 y^{2}}{30}}{x - \frac{2}{5} y}$

Step 1.13.6.1.4

Rewrite $21$ as $- 4$ plus $25$

$\frac{\frac{10 x^{2} + (- 4 + 25) (x y) - 10 y^{2}}{30}}{x - \frac{2}{5} y}$

Step 1.13.6.1.5

Apply the distributive property.

$\frac{\frac{10 x^{2} - 4 (x y) + 25 (x y) - 10 y^{2}}{30}}{x - \frac{2}{5} y}$

Step 1.13.6.1.6

Move parentheses.

$\frac{\frac{10 x^{2} - 4 x y + 25 x y - 10 y^{2}}{30}}{x - \frac{2}{5} y}$

Step 1.13.6.2

Factor out the greatest common factor from each group.

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Step 1.13.6.2.1

Group the first two terms and the last two terms.

$\frac{\frac{(10 x^{2} - 4 x y) + 25 x y - 10 y^{2}}{30}}{x - \frac{2}{5} y}$

Step 1.13.6.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{\frac{2 x (5 x - 2 y) + 5 y (5 x - 2 y)}{30}}{x - \frac{2}{5} y}$

Step 1.13.6.3

Factor the polynomial by factoring out the greatest common factor, $5 x - 2 y$ .

$\frac{\frac{(5 x - 2 y) (2 x + 5 y)}{30}}{x - \frac{2}{5} y}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\frac{(5 x - 2 y) (2 x + 5 y)}{30}}{x - \frac{y \cdot 2}{5}}$

Step 2.2

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

$\frac{\frac{(5 x - 2 y) (2 x + 5 y)}{30}}{x - \frac{2 \cdot y}{5}}$

Step 2.3

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

$\frac{\frac{(5 x - 2 y) (2 x + 5 y)}{30}}{x \cdot \frac{5}{5} - \frac{2 y}{5}}$

Step 2.4

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

$\frac{\frac{(5 x - 2 y) (2 x + 5 y)}{30}}{\frac{x \cdot 5}{5} - \frac{2 y}{5}}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{\frac{(5 x - 2 y) (2 x + 5 y)}{30}}{\frac{x \cdot 5 - 2 y}{5}}$

Step 2.6

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

$\frac{\frac{(5 x - 2 y) (2 x + 5 y)}{30}}{\frac{5 x - 2 y}{5}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{(5 x - 2 y) (2 x + 5 y)}{30} \cdot \frac{5}{5 x - 2 y}$

Step 4

Cancel the common factor of $5 x - 2 y$ .

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Step 4.1

Cancel the common factor.

$\frac{(5 x - 2 y) (2 x + 5 y)}{30} \cdot \frac{5}{5 x - 2 y}$

Step 4.2

Rewrite the expression.

$\frac{2 x + 5 y}{30} \cdot 5$

Step 5

Cancel the common factor of $5$ .

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Step 5.1

Factor $5$ out of $30$ .

$\frac{2 x + 5 y}{5 (6)} \cdot 5$

Step 5.2

Cancel the common factor.

$\frac{2 x + 5 y}{5 \cdot 6} \cdot 5$

Step 5.3

Rewrite the expression.

$\frac{2 x + 5 y}{6}$

Step 6

Split the fraction $\frac{2 x + 5 y}{6}$ into two fractions.

$\frac{2 x}{6} + \frac{5 y}{6}$

Step 7

Cancel the common factor of $2$ and $6$ .

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Step 7.1

Factor $2$ out of $2 x$ .

$\frac{2 (x)}{6} + \frac{5 y}{6}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $2$ out of $6$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

$\frac{x}{3} + \frac{5 y}{6}$