Basic Math Examples

$- 2 r + \frac{1}{3} - \frac{5}{2} r = - \frac{25}{6}$

Step 1

Simplify $- 2 r + \frac{1}{3} - \frac{5}{2} r$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$- 2 r + \frac{1}{3} - \frac{r \cdot 5}{2} = - \frac{25}{6}$

Step 1.1.2

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

$- 2 r + \frac{1}{3} - \frac{5 r}{2} = - \frac{25}{6}$

Step 1.2

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

$- 2 r \cdot \frac{2}{2} - \frac{5 r}{2} + \frac{1}{3} = - \frac{25}{6}$

Step 1.3

Simplify terms.

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Step 1.3.1

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

$\frac{- 2 r \cdot 2}{2} - \frac{5 r}{2} + \frac{1}{3} = - \frac{25}{6}$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{- 2 r \cdot 2 - 5 r}{2} + \frac{1}{3} = - \frac{25}{6}$

Step 1.4

Simplify each term.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Factor $r$ out of $- 2 r \cdot 2 - 5 r$ .

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Step 1.4.1.1.1

Factor $r$ out of $- 2 r \cdot 2$ .

$\frac{r (- 2 \cdot 2) - 5 r}{2} + \frac{1}{3} = - \frac{25}{6}$

Step 1.4.1.1.2

Factor $r$ out of $- 5 r$ .

$\frac{r (- 2 \cdot 2) + r \cdot - 5}{2} + \frac{1}{3} = - \frac{25}{6}$

Step 1.4.1.1.3

Factor $r$ out of $r (- 2 \cdot 2) + r \cdot - 5$ .

$\frac{r (- 2 \cdot 2 - 5)}{2} + \frac{1}{3} = - \frac{25}{6}$

Step 1.4.1.2

Multiply $- 2$ by $2$ .

$\frac{r (- 4 - 5)}{2} + \frac{1}{3} = - \frac{25}{6}$

Step 1.4.1.3

Subtract $5$ from $- 4$ .

$\frac{r \cdot - 9}{2} + \frac{1}{3} = - \frac{25}{6}$

Step 1.4.2

Move $- 9$ to the left of $r$ .

$\frac{- 9 \cdot r}{2} + \frac{1}{3} = - \frac{25}{6}$

Step 1.4.3

Move the negative in front of the fraction.

$- \frac{9 r}{2} + \frac{1}{3} = - \frac{25}{6}$

Step 2

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

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Step 2.1

Subtract $\frac{1}{3}$ from both sides of the equation.

$- \frac{9 r}{2} = - \frac{25}{6} - \frac{1}{3}$

Step 2.2

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

$- \frac{9 r}{2} = - \frac{25}{6} - \frac{1}{3} \cdot \frac{2}{2}$

Step 2.3

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

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Step 2.3.1

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

$- \frac{9 r}{2} = - \frac{25}{6} - \frac{2}{3 \cdot 2}$

Step 2.3.2

Multiply $3$ by $2$ .

$- \frac{9 r}{2} = - \frac{25}{6} - \frac{2}{6}$

Step 2.4

Combine the numerators over the common denominator.

$- \frac{9 r}{2} = \frac{- 25 - 2}{6}$

Step 2.5

Subtract $2$ from $- 25$ .

$- \frac{9 r}{2} = \frac{- 27}{6}$

Step 2.6

Cancel the common factor of $- 27$ and $6$ .

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Step 2.6.1

Factor $3$ out of $- 27$ .

$- \frac{9 r}{2} = \frac{3 (- 9)}{6}$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $3$ out of $6$ .

$- \frac{9 r}{2} = \frac{3 \cdot - 9}{3 \cdot 2}$

Step 2.6.2.2

Cancel the common factor.

$- \frac{9 r}{2} = \frac{3 \cdot - 9}{3 \cdot 2}$

Step 2.6.2.3

Rewrite the expression.

$- \frac{9 r}{2} = \frac{- 9}{2}$

Step 2.7

Move the negative in front of the fraction.

$- \frac{9 r}{2} = - \frac{9}{2}$

Step 3

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

$- 9 r = - 9$

Step 4

Divide each term in $- 9 r = - 9$ by $- 9$ and simplify.

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Step 4.1

Divide each term in $- 9 r = - 9$ by $- 9$ .

$\frac{- 9 r}{- 9} = \frac{- 9}{- 9}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 9$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 9 r}{- 9} = \frac{- 9}{- 9}$

Step 4.2.1.2

Divide $r$ by $1$ .

$r = \frac{- 9}{- 9}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 9$ by $- 9$ .

$r = 1$