Finite Math Examples

$\frac{1}{3} \cdot (p + 2) = \frac{3}{6} \cdot (p - 1)$

Step 1

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

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Step 1.1

Rewrite.

$0 + 0 + \frac{1}{3} \cdot (p + 2) = \frac{3}{6} \cdot (p - 1)$

Step 1.2

Simplify by adding zeros.

$\frac{1}{3} \cdot (p + 2) = \frac{3}{6} \cdot (p - 1)$

Step 1.3

Apply the distributive property.

$\frac{1}{3} p + \frac{1}{3} \cdot 2 = \frac{3}{6} \cdot (p - 1)$

Step 1.4

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

$\frac{p}{3} + \frac{1}{3} \cdot 2 = \frac{3}{6} \cdot (p - 1)$

Step 1.5

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

$\frac{p}{3} + \frac{2}{3} = \frac{3}{6} \cdot (p - 1)$

Step 2

Simplify $\frac{3}{6} \cdot (p - 1)$ .

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Step 2.1

Cancel the common factor of $3$ and $6$ .

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Step 2.1.1

Factor $3$ out of $3$ .

$\frac{p}{3} + \frac{2}{3} = \frac{3 (1)}{6} \cdot (p - 1)$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $3$ out of $6$ .

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

Step 2.1.2.2

Cancel the common factor.

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

Step 2.1.2.3

Rewrite the expression.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

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

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

Step 2.4

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

$\frac{p}{3} + \frac{2}{3} = \frac{p}{2} + \frac{- 1}{2}$

Step 2.5

Move the negative in front of the fraction.

$\frac{p}{3} + \frac{2}{3} = \frac{p}{2} - \frac{1}{2}$

Step 3

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

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Step 3.1

Subtract $\frac{p}{2}$ from both sides of the equation.

$\frac{p}{3} + \frac{2}{3} - \frac{p}{2} = - \frac{1}{2}$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Multiply $3$ by $2$ .

$\frac{p \cdot 2}{6} - \frac{p}{2} \cdot \frac{3}{3} + \frac{2}{3} = - \frac{1}{2}$

Step 3.4.3

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

$\frac{p \cdot 2}{6} - \frac{p \cdot 3}{2 \cdot 3} + \frac{2}{3} = - \frac{1}{2}$

Step 3.4.4

Multiply $2$ by $3$ .

$\frac{p \cdot 2}{6} - \frac{p \cdot 3}{6} + \frac{2}{3} = - \frac{1}{2}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{p \cdot 2 - p \cdot 3}{6} + \frac{2}{3} = - \frac{1}{2}$

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 $p \cdot 2 - p \cdot 3$ .

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Step 3.6.1.1.1

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

$\frac{p (2) - p \cdot 3}{6} + \frac{2}{3} = - \frac{1}{2}$

Step 3.6.1.1.2

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

$\frac{p (2) + p (- 1 \cdot 3)}{6} + \frac{2}{3} = - \frac{1}{2}$

Step 3.6.1.1.3

Factor $p$ out of $p (2) + p (- 1 \cdot 3)$ .

$\frac{p (2 - 1 \cdot 3)}{6} + \frac{2}{3} = - \frac{1}{2}$

Step 3.6.1.2

Multiply $- 1$ by $3$ .

$\frac{p (2 - 3)}{6} + \frac{2}{3} = - \frac{1}{2}$

Step 3.6.1.3

Subtract $3$ from $2$ .

$\frac{p \cdot - 1}{6} + \frac{2}{3} = - \frac{1}{2}$

Step 3.6.2

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

$\frac{- 1 \cdot p}{6} + \frac{2}{3} = - \frac{1}{2}$

Step 3.6.3

Move the negative in front of the fraction.

$- \frac{p}{6} + \frac{2}{3} = - \frac{1}{2}$

Step 4

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

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Step 4.1

Subtract $\frac{2}{3}$ from both sides of the equation.

$- \frac{p}{6} = - \frac{1}{2} - \frac{2}{3}$

Step 4.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{p}{6} = - \frac{1}{2} \cdot \frac{3}{3} - \frac{2}{3}$

Step 4.3

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

$- \frac{p}{6} = - \frac{1}{2} \cdot \frac{3}{3} - \frac{2}{3} \cdot \frac{2}{2}$

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

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

Step 4.4.2

Multiply $2$ by $3$ .

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

Step 4.4.3

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

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

Step 4.4.4

Multiply $3$ by $2$ .

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 2$ by $2$ .

$- \frac{p}{6} = \frac{- 3 - 4}{6}$

Step 4.6.2

Subtract $4$ from $- 3$ .

$- \frac{p}{6} = \frac{- 7}{6}$

Step 4.7

Move the negative in front of the fraction.

$- \frac{p}{6} = - \frac{7}{6}$

Step 5

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- p = - 7$

Step 6

Divide each term in $- p = - 7$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- p = - 7$ by $- 1$ .

$\frac{- p}{- 1} = \frac{- 7}{- 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{p}{1} = \frac{- 7}{- 1}$

Step 6.2.2

Divide $p$ by $1$ .

$p = \frac{- 7}{- 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $- 7$ by $- 1$ .

$p = 7$