Pre-Algebra Examples

$- \frac{2}{7} r + 2 r = \frac{1}{2} r + \frac{17}{2}$

Step 1

Simplify $- \frac{2}{7} r + 2 r$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $r$ and $\frac{2}{7}$ .

$- \frac{r \cdot 2}{7} + 2 r = \frac{1}{2} r + \frac{17}{2}$

Step 1.1.2

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

$- \frac{2 r}{7} + 2 r = \frac{1}{2} r + \frac{17}{2}$

Step 1.2

To write $2 r$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- \frac{2 r}{7} + 2 r \cdot \frac{7}{7} = \frac{1}{2} r + \frac{17}{2}$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $2 r$ and $\frac{7}{7}$ .

$- \frac{2 r}{7} + \frac{2 r \cdot 7}{7} = \frac{1}{2} r + \frac{17}{2}$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{- 2 r + 2 r \cdot 7}{7} = \frac{1}{2} r + \frac{17}{2}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Factor $2 r$ out of $- 2 r + 2 r \cdot 7$ .

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Step 1.4.1.1

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

$\frac{2 r (- 1) + 2 r \cdot 7}{7} = \frac{1}{2} r + \frac{17}{2}$

Step 1.4.1.2

Factor $2 r$ out of $2 r \cdot 7$ .

$\frac{2 r (- 1) + 2 r (7)}{7} = \frac{1}{2} r + \frac{17}{2}$

Step 1.4.1.3

Factor $2 r$ out of $2 r (- 1) + 2 r (7)$ .

$\frac{2 r (- 1 + 7)}{7} = \frac{1}{2} r + \frac{17}{2}$

Step 1.4.2

Add $- 1$ and $7$ .

$\frac{2 r \cdot 6}{7} = \frac{1}{2} r + \frac{17}{2}$

Step 1.4.3

Multiply $6$ by $2$ .

$\frac{12 r}{7} = \frac{1}{2} r + \frac{17}{2}$

Step 2

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

$\frac{12 r}{7} = \frac{r}{2} + \frac{17}{2}$

Step 3

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

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Step 3.1

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

$\frac{12 r}{7} - \frac{r}{2} = \frac{17}{2}$

Step 3.2

To write $\frac{12 r}{7}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{12 r}{7} \cdot \frac{2}{2} - \frac{r}{2} = \frac{17}{2}$

Step 3.3

To write $- \frac{r}{2}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{12 r}{7} \cdot \frac{2}{2} - \frac{r}{2} \cdot \frac{7}{7} = \frac{17}{2}$

Step 3.4

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{12 r}{7}$ by $\frac{2}{2}$ .

$\frac{12 r \cdot 2}{7 \cdot 2} - \frac{r}{2} \cdot \frac{7}{7} = \frac{17}{2}$

Step 3.4.2

Multiply $7$ by $2$ .

$\frac{12 r \cdot 2}{14} - \frac{r}{2} \cdot \frac{7}{7} = \frac{17}{2}$

Step 3.4.3

Multiply $\frac{r}{2}$ by $\frac{7}{7}$ .

$\frac{12 r \cdot 2}{14} - \frac{r \cdot 7}{2 \cdot 7} = \frac{17}{2}$

Step 3.4.4

Multiply $2$ by $7$ .

$\frac{12 r \cdot 2}{14} - \frac{r \cdot 7}{14} = \frac{17}{2}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{12 r \cdot 2 - r \cdot 7}{14} = \frac{17}{2}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Factor $r$ out of $12 r \cdot 2 - r \cdot 7$ .

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Step 3.6.1.1

Factor $r$ out of $12 r \cdot 2$ .

$\frac{r (12 \cdot 2) - r \cdot 7}{14} = \frac{17}{2}$

Step 3.6.1.2

Factor $r$ out of $- r \cdot 7$ .

$\frac{r (12 \cdot 2) + r (- 1 \cdot 7)}{14} = \frac{17}{2}$

Step 3.6.1.3

Factor $r$ out of $r (12 \cdot 2) + r (- 1 \cdot 7)$ .

$\frac{r (12 \cdot 2 - 1 \cdot 7)}{14} = \frac{17}{2}$

Step 3.6.2

Multiply $12$ by $2$ .

$\frac{r (24 - 1 \cdot 7)}{14} = \frac{17}{2}$

Step 3.6.3

Multiply $- 1$ by $7$ .

$\frac{r (24 - 7)}{14} = \frac{17}{2}$

Step 3.6.4

Subtract $7$ from $24$ .

$\frac{r \cdot 17}{14} = \frac{17}{2}$

Step 3.7

Move $17$ to the left of $r$ .

$\frac{17 r}{14} = \frac{17}{2}$

Step 4

Multiply both sides of the equation by $\frac{14}{17}$ .

$\frac{14}{17} \cdot \frac{17 r}{14} = \frac{14}{17} \cdot \frac{17}{2}$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $\frac{14}{17} \cdot \frac{17 r}{14}$ .

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Step 5.1.1.1

Cancel the common factor of $14$ .

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Step 5.1.1.1.1

Cancel the common factor.

$\frac{14}{17} \cdot \frac{17 r}{14} = \frac{14}{17} \cdot \frac{17}{2}$

Step 5.1.1.1.2

Rewrite the expression.

$\frac{1}{17} (17 r) = \frac{14}{17} \cdot \frac{17}{2}$

Step 5.1.1.2

Cancel the common factor of $17$ .

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Step 5.1.1.2.1

Factor $17$ out of $17 r$ .

$\frac{1}{17} (17 (r)) = \frac{14}{17} \cdot \frac{17}{2}$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{1}{17} (17 r) = \frac{14}{17} \cdot \frac{17}{2}$

Step 5.1.1.2.3

Rewrite the expression.

$r = \frac{14}{17} \cdot \frac{17}{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{14}{17} \cdot \frac{17}{2}$ .

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Factor $2$ out of $14$ .

$r = \frac{2 (7)}{17} \cdot \frac{17}{2}$

Step 5.2.1.1.2

Cancel the common factor.

$r = \frac{2 \cdot 7}{17} \cdot \frac{17}{2}$

Step 5.2.1.1.3

Rewrite the expression.

$r = \frac{7}{17} \cdot 17$

Step 5.2.1.2

Cancel the common factor of $17$ .

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Step 5.2.1.2.1

Cancel the common factor.

$r = \frac{7}{17} \cdot 17$

Step 5.2.1.2.2

Rewrite the expression.

$r = 7$