Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

Combine $y$ and $\frac{5}{6}$ .

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

Step 1.1.3

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

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

Step 1.2

Simplify each term.

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Step 1.2.1

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

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

Step 1.2.2

Combine $\frac{3}{5}$ and $y$ .

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

Step 2

Expand $(\frac{x}{3} - \frac{5 y}{6}) (- \frac{x}{2} + \frac{3 y}{5})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.1.2

Multiply $- \frac{x}{3} \cdot \frac{x}{2}$ .

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Step 3.1.2.1

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

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

Step 3.1.2.2

Raise $x$ to the power of $1$ .

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

Step 3.1.2.3

Raise $x$ to the power of $1$ .

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

Step 3.1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.1.2.5

Add $1$ and $1$ .

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

Step 3.1.2.6

Multiply $2$ by $3$ .

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

Step 3.1.3

Cancel the common factor of $3$ .

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Step 3.1.3.1

Factor $3$ out of $3 y$ .

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

Step 3.1.3.2

Cancel the common factor.

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

Step 3.1.3.3

Rewrite the expression.

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

Step 3.1.4

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

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

Step 3.1.5

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

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Step 3.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.5.2

Multiply $\frac{5 y}{6}$ by $1$ .

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

Step 3.1.5.3

Multiply $\frac{5 y}{6}$ by $\frac{x}{2}$ .

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

Step 3.1.5.4

Multiply $6$ by $2$ .

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

Step 3.1.6

Cancel the common factor of $5$ .

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Step 3.1.6.1

Move the leading negative in $- \frac{5 y}{6}$ into the numerator.

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

Step 3.1.6.2

Factor $5$ out of $- 5 y$ .

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

Step 3.1.6.3

Cancel the common factor.

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

Step 3.1.6.4

Rewrite the expression.

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

Step 3.1.7

Cancel the common factor of $3$ .

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Step 3.1.7.1

Factor $3$ out of $6$ .

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

Step 3.1.7.2

Factor $3$ out of $3 y$ .

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

Step 3.1.7.3

Cancel the common factor.

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

Step 3.1.7.4

Rewrite the expression.

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

Step 3.1.8

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

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

Step 3.1.9

Raise $y$ to the power of $1$ .

$- \frac{x^{2}}{6} + \frac{x y}{5} + \frac{5 y x}{12} + \frac{- (y^{1} y)}{2}$

Step 3.1.10

Raise $y$ to the power of $1$ .

$- \frac{x^{2}}{6} + \frac{x y}{5} + \frac{5 y x}{12} + \frac{- (y^{1} y^{1})}{2}$

Step 3.1.11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{x^{2}}{6} + \frac{x y}{5} + \frac{5 y x}{12} + \frac{- y^{1 + 1}}{2}$

Step 3.1.12

Add $1$ and $1$ .

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

Step 3.1.13

Move the negative in front of the fraction.

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

Step 3.2

Add $\frac{x y}{5}$ and $\frac{5 y x}{12}$ .

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Step 3.2.1

Move $y$ .

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

Step 3.2.2

To write $\frac{x y}{5}$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

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

Step 3.2.3

To write $\frac{5 x y}{12}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 3.2.4

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.4.1

Multiply $\frac{x y}{5}$ by $\frac{12}{12}$ .

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

Step 3.2.4.2

Multiply $5$ by $12$ .

$- \frac{x^{2}}{6} + \frac{x y \cdot 12}{60} + \frac{5 x y}{12} \cdot \frac{5}{5} - \frac{y^{2}}{2}$

Step 3.2.4.3

Multiply $\frac{5 x y}{12}$ by $\frac{5}{5}$ .

$- \frac{x^{2}}{6} + \frac{x y \cdot 12}{60} + \frac{5 x y \cdot 5}{12 \cdot 5} - \frac{y^{2}}{2}$

Step 3.2.4.4

Multiply $12$ by $5$ .

$- \frac{x^{2}}{6} + \frac{x y \cdot 12}{60} + \frac{5 x y \cdot 5}{60} - \frac{y^{2}}{2}$

Step 3.2.5

Combine the numerators over the common denominator.

$- \frac{x^{2}}{6} + \frac{x y \cdot 12 + 5 x y \cdot 5}{60} - \frac{y^{2}}{2}$

Step 3.3

To write $- \frac{x^{2}}{6}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$- \frac{x^{2}}{6} \cdot \frac{10}{10} + \frac{x y \cdot 12 + 5 x y \cdot 5}{60} - \frac{y^{2}}{2}$

Step 3.4

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{x^{2}}{6}$ by $\frac{10}{10}$ .

$- \frac{x^{2} \cdot 10}{6 \cdot 10} + \frac{x y \cdot 12 + 5 x y \cdot 5}{60} - \frac{y^{2}}{2}$

Step 3.4.2

Multiply $6$ by $10$ .

$- \frac{x^{2} \cdot 10}{60} + \frac{x y \cdot 12 + 5 x y \cdot 5}{60} - \frac{y^{2}}{2}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{- x^{2} \cdot 10 + x y \cdot 12 + 5 x y \cdot 5}{60} - \frac{y^{2}}{2}$

Step 3.6

To write $- \frac{y^{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{30}{30}$ .

$\frac{- x^{2} \cdot 10 + x y \cdot 12 + 5 x y \cdot 5}{60} - \frac{y^{2}}{2} \cdot \frac{30}{30}$

Step 3.7

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.7.1

Multiply $\frac{y^{2}}{2}$ by $\frac{30}{30}$ .

$\frac{- x^{2} \cdot 10 + x y \cdot 12 + 5 x y \cdot 5}{60} - \frac{y^{2} \cdot 30}{2 \cdot 30}$

Step 3.7.2

Multiply $2$ by $30$ .

$\frac{- x^{2} \cdot 10 + x y \cdot 12 + 5 x y \cdot 5}{60} - \frac{y^{2} \cdot 30}{60}$

Step 3.8

Combine the numerators over the common denominator.

$\frac{- x^{2} \cdot 10 + x y \cdot 12 + 5 x y \cdot 5 - y^{2} \cdot 30}{60}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $10$ by $- 1$ .

$\frac{- 10 x^{2} + x y \cdot 12 + 5 x y \cdot 5 - y^{2} \cdot 30}{60}$

Step 4.2

Move $12$ to the left of $x y$ .

$\frac{- 10 x^{2} + 12 \cdot (x y) + 5 x y \cdot 5 - y^{2} \cdot 30}{60}$

Step 4.3

Multiply $5$ by $5$ .

$\frac{- 10 x^{2} + 12 x y + 25 x y - y^{2} \cdot 30}{60}$

Step 4.4

Multiply $30$ by $- 1$ .

$\frac{- 10 x^{2} + 12 x y + 25 x y - 30 y^{2}}{60}$

Step 4.5

Add $12 x y$ and $25 x y$ .

$\frac{- 10 x^{2} + 37 x y - 30 y^{2}}{60}$

Step 4.6

Factor by grouping.

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Step 4.6.1

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 $a \cdot c = - 10 \cdot - 30 = 300$ and whose sum is $b = 37$ .

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Step 4.6.1.1

Reorder terms.

$\frac{- 10 x^{2} - 30 y^{2} + 37 x y}{60}$

Step 4.6.1.2

Reorder $- 30 y^{2}$ and $37 x y$ .

$\frac{- 10 x^{2} + 37 x y - 30 y^{2}}{60}$

Step 4.6.1.3

Factor $37$ out of $37 x y$ .

$\frac{- 10 x^{2} + 37 (x y) - 30 y^{2}}{60}$

Step 4.6.1.4

Rewrite $37$ as $12$ plus $25$

$\frac{- 10 x^{2} + (12 + 25) (x y) - 30 y^{2}}{60}$

Step 4.6.1.5

Apply the distributive property.

$\frac{- 10 x^{2} + 12 (x y) + 25 (x y) - 30 y^{2}}{60}$

Step 4.6.1.6

Move parentheses.

$\frac{- 10 x^{2} + 12 x y + 25 x y - 30 y^{2}}{60}$

Step 4.6.2

Factor out the greatest common factor from each group.

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Step 4.6.2.1

Group the first two terms and the last two terms.

$\frac{(- 10 x^{2} + 12 x y) + 25 x y - 30 y^{2}}{60}$

Step 4.6.2.2

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

$\frac{2 x (- 5 x + 6 y) - 5 y (- 5 x + 6 y)}{60}$

Step 4.6.3

Factor the polynomial by factoring out the greatest common factor, $- 5 x + 6 y$ .

$\frac{(- 5 x + 6 y) (2 x - 5 y)}{60}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $- 1$ out of $- 5 x$ .

$\frac{(- (5 x) + 6 y) (2 x - 5 y)}{60}$

Step 5.2

Factor $- 1$ out of $6 y$ .

$\frac{(- (5 x) - (- 6 y)) (2 x - 5 y)}{60}$

Step 5.3

Factor $- 1$ out of $- (5 x) - (- 6 y)$ .

$\frac{- (5 x - 6 y) (2 x - 5 y)}{60}$

Step 5.4

Simplify the expression.

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Step 5.4.1

Rewrite $- (5 x - 6 y)$ as $- 1 (5 x - 6 y)$ .

$\frac{- 1 (5 x - 6 y) (2 x - 5 y)}{60}$

Step 5.4.2

Move the negative in front of the fraction.

$- \frac{(5 x - 6 y) (2 x - 5 y)}{60}$