Algebra Examples

$4 + \frac{2}{3} (p + 1) = 3 + \frac{3}{4} (p + 2)$

Step 1

Simplify $4 + \frac{2}{3} \cdot (p + 1)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$4 + \frac{2}{3} p + \frac{2}{3} \cdot 1 = 3 + \frac{3}{4} \cdot (p + 2)$

Step 1.1.2

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

$4 + \frac{2 p}{3} + \frac{2}{3} \cdot 1 = 3 + \frac{3}{4} \cdot (p + 2)$

Step 1.1.3

Multiply $\frac{2}{3}$ by $1$ .

$4 + \frac{2 p}{3} + \frac{2}{3} = 3 + \frac{3}{4} \cdot (p + 2)$

Step 1.2

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

$\frac{2 p}{3} + 4 \cdot \frac{3}{3} + \frac{2}{3} = 3 + \frac{3}{4} \cdot (p + 2)$

Step 1.3

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

$\frac{2 p}{3} + \frac{4 \cdot 3}{3} + \frac{2}{3} = 3 + \frac{3}{4} \cdot (p + 2)$

Step 1.4

Combine the numerators over the common denominator.

$\frac{2 p}{3} + \frac{4 \cdot 3 + 2}{3} = 3 + \frac{3}{4} \cdot (p + 2)$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $4$ by $3$ .

$\frac{2 p}{3} + \frac{12 + 2}{3} = 3 + \frac{3}{4} \cdot (p + 2)$

Step 1.5.2

Add $12$ and $2$ .

$\frac{2 p}{3} + \frac{14}{3} = 3 + \frac{3}{4} \cdot (p + 2)$

Step 2

Simplify $3 + \frac{3}{4} \cdot (p + 2)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$\frac{2 p}{3} + \frac{14}{3} = 3 + \frac{3}{4} p + \frac{3}{4} \cdot 2$

Step 2.1.2

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

$\frac{2 p}{3} + \frac{14}{3} = 3 + \frac{3 p}{4} + \frac{3}{4} \cdot 2$

Step 2.1.3

Cancel the common factor of $2$ .

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Step 2.1.3.1

Factor $2$ out of $4$ .

$\frac{2 p}{3} + \frac{14}{3} = 3 + \frac{3 p}{4} + \frac{3}{2 (2)} \cdot 2$

Step 2.1.3.2

Cancel the common factor.

$\frac{2 p}{3} + \frac{14}{3} = 3 + \frac{3 p}{4} + \frac{3}{2 \cdot 2} \cdot 2$

Step 2.1.3.3

Rewrite the expression.

$\frac{2 p}{3} + \frac{14}{3} = 3 + \frac{3 p}{4} + \frac{3}{2}$

Step 2.2

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

$\frac{2 p}{3} + \frac{14}{3} = \frac{3 p}{4} + 3 \cdot \frac{2}{2} + \frac{3}{2}$

Step 2.3

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

$\frac{2 p}{3} + \frac{14}{3} = \frac{3 p}{4} + \frac{3 \cdot 2}{2} + \frac{3}{2}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{2 p}{3} + \frac{14}{3} = \frac{3 p}{4} + \frac{3 \cdot 2 + 3}{2}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $3$ by $2$ .

$\frac{2 p}{3} + \frac{14}{3} = \frac{3 p}{4} + \frac{6 + 3}{2}$

Step 2.5.2

Add $6$ and $3$ .

$\frac{2 p}{3} + \frac{14}{3} = \frac{3 p}{4} + \frac{9}{2}$

Step 3

Move all terms containing $p$ to the left side of the equation.

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Step 3.1

Subtract $\frac{3 p}{4}$ from both sides of the equation.

$\frac{2 p}{3} + \frac{14}{3} - \frac{3 p}{4} = \frac{9}{2}$

Step 3.2

To write $\frac{2 p}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{2 p}{3} \cdot \frac{4}{4} - \frac{3 p}{4} + \frac{14}{3} = \frac{9}{2}$

Step 3.3

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

$\frac{2 p}{3} \cdot \frac{4}{4} - \frac{3 p}{4} \cdot \frac{3}{3} + \frac{14}{3} = \frac{9}{2}$

Step 3.4

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

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Step 3.4.1

Multiply $\frac{2 p}{3}$ by $\frac{4}{4}$ .

$\frac{2 p \cdot 4}{3 \cdot 4} - \frac{3 p}{4} \cdot \frac{3}{3} + \frac{14}{3} = \frac{9}{2}$

Step 3.4.2

Multiply $3$ by $4$ .

$\frac{2 p \cdot 4}{12} - \frac{3 p}{4} \cdot \frac{3}{3} + \frac{14}{3} = \frac{9}{2}$

Step 3.4.3

Multiply $\frac{3 p}{4}$ by $\frac{3}{3}$ .

$\frac{2 p \cdot 4}{12} - \frac{3 p \cdot 3}{4 \cdot 3} + \frac{14}{3} = \frac{9}{2}$

Step 3.4.4

Multiply $4$ by $3$ .

$\frac{2 p \cdot 4}{12} - \frac{3 p \cdot 3}{12} + \frac{14}{3} = \frac{9}{2}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{2 p \cdot 4 - 3 p \cdot 3}{12} + \frac{14}{3} = \frac{9}{2}$

Step 3.6

Simplify each term.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Factor $p$ out of $2 p \cdot 4 - 3 p \cdot 3$ .

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Step 3.6.1.1.1

Factor $p$ out of $2 p \cdot 4$ .

$\frac{p (2 \cdot 4) - 3 p \cdot 3}{12} + \frac{14}{3} = \frac{9}{2}$

Step 3.6.1.1.2

Factor $p$ out of $- 3 p \cdot 3$ .

$\frac{p (2 \cdot 4) + p (- 3 \cdot 3)}{12} + \frac{14}{3} = \frac{9}{2}$

Step 3.6.1.1.3

Factor $p$ out of $p (2 \cdot 4) + p (- 3 \cdot 3)$ .

$\frac{p (2 \cdot 4 - 3 \cdot 3)}{12} + \frac{14}{3} = \frac{9}{2}$

Step 3.6.1.2

Multiply $2$ by $4$ .

$\frac{p (8 - 3 \cdot 3)}{12} + \frac{14}{3} = \frac{9}{2}$

Step 3.6.1.3

Multiply $- 3$ by $3$ .

$\frac{p (8 - 9)}{12} + \frac{14}{3} = \frac{9}{2}$

Step 3.6.1.4

Subtract $9$ from $8$ .

$\frac{p \cdot - 1}{12} + \frac{14}{3} = \frac{9}{2}$

Step 3.6.2

Move $- 1$ to the left of $p$ .

$\frac{- 1 \cdot p}{12} + \frac{14}{3} = \frac{9}{2}$

Step 3.6.3

Move the negative in front of the fraction.

$- \frac{p}{12} + \frac{14}{3} = \frac{9}{2}$

Step 4

Move all terms not containing $p$ to the right side of the equation.

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Step 4.1

Subtract $\frac{14}{3}$ from both sides of the equation.

$- \frac{p}{12} = \frac{9}{2} - \frac{14}{3}$

Step 4.2

To write $\frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{p}{12} = \frac{9}{2} \cdot \frac{3}{3} - \frac{14}{3}$

Step 4.3

To write $- \frac{14}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{p}{12} = \frac{9}{2} \cdot \frac{3}{3} - \frac{14}{3} \cdot \frac{2}{2}$

Step 4.4

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

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Step 4.4.1

Multiply $\frac{9}{2}$ by $\frac{3}{3}$ .

$- \frac{p}{12} = \frac{9 \cdot 3}{2 \cdot 3} - \frac{14}{3} \cdot \frac{2}{2}$

Step 4.4.2

Multiply $2$ by $3$ .

$- \frac{p}{12} = \frac{9 \cdot 3}{6} - \frac{14}{3} \cdot \frac{2}{2}$

Step 4.4.3

Multiply $\frac{14}{3}$ by $\frac{2}{2}$ .

$- \frac{p}{12} = \frac{9 \cdot 3}{6} - \frac{14 \cdot 2}{3 \cdot 2}$

Step 4.4.4

Multiply $3$ by $2$ .

$- \frac{p}{12} = \frac{9 \cdot 3}{6} - \frac{14 \cdot 2}{6}$

Step 4.5

Combine the numerators over the common denominator.

$- \frac{p}{12} = \frac{9 \cdot 3 - 14 \cdot 2}{6}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $9$ by $3$ .

$- \frac{p}{12} = \frac{27 - 14 \cdot 2}{6}$

Step 4.6.2

Multiply $- 14$ by $2$ .

$- \frac{p}{12} = \frac{27 - 28}{6}$

Step 4.6.3

Subtract $28$ from $27$ .

$- \frac{p}{12} = \frac{- 1}{6}$

Step 4.7

Move the negative in front of the fraction.

$- \frac{p}{12} = - \frac{1}{6}$

Step 5

Multiply both sides of the equation by $- 12$ .

$- 12 (- \frac{p}{12}) = - 12 (- \frac{1}{6})$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $- 12 (- \frac{p}{12})$ .

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Step 6.1.1.1

Cancel the common factor of $12$ .

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Step 6.1.1.1.1

Move the leading negative in $- \frac{p}{12}$ into the numerator.

$- 12 \frac{- p}{12} = - 12 (- \frac{1}{6})$

Step 6.1.1.1.2

Factor $12$ out of $- 12$ .

$12 (- 1) \frac{- p}{12} = - 12 (- \frac{1}{6})$

Step 6.1.1.1.3

Cancel the common factor.

$12 \cdot - 1 \frac{- p}{12} = - 12 (- \frac{1}{6})$

Step 6.1.1.1.4

Rewrite the expression.

$- - p = - 12 (- \frac{1}{6})$

Step 6.1.1.2

Multiply.

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Step 6.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 p = - 12 (- \frac{1}{6})$

Step 6.1.1.2.2

Multiply $p$ by $1$ .

$p = - 12 (- \frac{1}{6})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 12 (- \frac{1}{6})$ .

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Step 6.2.1.1

Cancel the common factor of $6$ .

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Step 6.2.1.1.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$p = - 12 (\frac{- 1}{6})$

Step 6.2.1.1.2

Factor $6$ out of $- 12$ .

$p = 6 (- 2) \frac{- 1}{6}$

Step 6.2.1.1.3

Cancel the common factor.

$p = 6 \cdot - 2 \frac{- 1}{6}$

Step 6.2.1.1.4

Rewrite the expression.

$p = - 2 \cdot - 1$

Step 6.2.1.2

Multiply $- 2$ by $- 1$ .

$p = 2$