Algebra Examples

$\frac{8}{9} (\frac{1}{2} v - 18) + 2 v = \frac{2}{3} v$

Step 1

Simplify $\frac{8}{9} \cdot (\frac{1}{2} v - 18) + 2 v$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$\frac{8}{9} \cdot (\frac{v}{2} - 18) + 2 v = \frac{2}{3} v$

Step 1.1.2

Apply the distributive property.

$\frac{8}{9} \cdot \frac{v}{2} + \frac{8}{9} \cdot - 18 + 2 v = \frac{2}{3} v$

Step 1.1.3

Cancel the common factor of $2$ .

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Step 1.1.3.1

Factor $2$ out of $8$ .

$\frac{2 (4)}{9} \cdot \frac{v}{2} + \frac{8}{9} \cdot - 18 + 2 v = \frac{2}{3} v$

Step 1.1.3.2

Cancel the common factor.

$\frac{2 \cdot 4}{9} \cdot \frac{v}{2} + \frac{8}{9} \cdot - 18 + 2 v = \frac{2}{3} v$

Step 1.1.3.3

Rewrite the expression.

$\frac{4}{9} v + \frac{8}{9} \cdot - 18 + 2 v = \frac{2}{3} v$

Step 1.1.4

Combine $\frac{4}{9}$ and $v$ .

$\frac{4 v}{9} + \frac{8}{9} \cdot - 18 + 2 v = \frac{2}{3} v$

Step 1.1.5

Cancel the common factor of $9$ .

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Step 1.1.5.1

Factor $9$ out of $- 18$ .

$\frac{4 v}{9} + \frac{8}{9} \cdot (9 (- 2)) + 2 v = \frac{2}{3} v$

Step 1.1.5.2

Cancel the common factor.

$\frac{4 v}{9} + \frac{8}{9} \cdot (9 \cdot - 2) + 2 v = \frac{2}{3} v$

Step 1.1.5.3

Rewrite the expression.

$\frac{4 v}{9} + 8 \cdot - 2 + 2 v = \frac{2}{3} v$

Step 1.1.6

Multiply $8$ by $- 2$ .

$\frac{4 v}{9} - 16 + 2 v = \frac{2}{3} v$

Step 1.2

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

$\frac{4 v}{9} + 2 v \cdot \frac{9}{9} - 16 = \frac{2}{3} v$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $2 v$ and $\frac{9}{9}$ .

$\frac{4 v}{9} + \frac{2 v \cdot 9}{9} - 16 = \frac{2}{3} v$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{4 v + 2 v \cdot 9}{9} - 16 = \frac{2}{3} v$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Factor $2 v$ out of $4 v + 2 v \cdot 9$ .

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Step 1.4.1.1

Factor $2 v$ out of $4 v$ .

$\frac{2 v (2) + 2 v \cdot 9}{9} - 16 = \frac{2}{3} v$

Step 1.4.1.2

Factor $2 v$ out of $2 v \cdot 9$ .

$\frac{2 v (2) + 2 v (9)}{9} - 16 = \frac{2}{3} v$

Step 1.4.1.3

Factor $2 v$ out of $2 v (2) + 2 v (9)$ .

$\frac{2 v (2 + 9)}{9} - 16 = \frac{2}{3} v$

Step 1.4.2

Add $2$ and $9$ .

$\frac{2 v \cdot 11}{9} - 16 = \frac{2}{3} v$

Step 1.4.3

Multiply $11$ by $2$ .

$\frac{22 v}{9} - 16 = \frac{2}{3} v$

Step 2

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

$\frac{22 v}{9} - 16 = \frac{2 v}{3}$

Step 3

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

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Step 3.1

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

$\frac{22 v}{9} - 16 - \frac{2 v}{3} = 0$

Step 3.2

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

$\frac{22 v}{9} - \frac{2 v}{3} \cdot \frac{3}{3} - 16 = 0$

Step 3.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

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

$\frac{22 v}{9} - \frac{2 v \cdot 3}{3 \cdot 3} - 16 = 0$

Step 3.3.2

Multiply $3$ by $3$ .

$\frac{22 v}{9} - \frac{2 v \cdot 3}{9} - 16 = 0$

Step 3.4

Combine the numerators over the common denominator.

$\frac{22 v - 2 v \cdot 3}{9} - 16 = 0$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Factor $2 v$ out of $22 v - 2 v \cdot 3$ .

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Step 3.5.1.1

Factor $2 v$ out of $22 v$ .

$\frac{2 v (11) - 2 v \cdot 3}{9} - 16 = 0$

Step 3.5.1.2

Factor $2 v$ out of $- 2 v \cdot 3$ .

$\frac{2 v (11) + 2 v (- 1 \cdot 3)}{9} - 16 = 0$

Step 3.5.1.3

Factor $2 v$ out of $2 v (11) + 2 v (- 1 \cdot 3)$ .

$\frac{2 v (11 - 1 \cdot 3)}{9} - 16 = 0$

Step 3.5.2

Multiply $- 1$ by $3$ .

$\frac{2 v (11 - 3)}{9} - 16 = 0$

Step 3.5.3

Subtract $3$ from $11$ .

$\frac{2 v \cdot 8}{9} - 16 = 0$

Step 3.5.4

Multiply $8$ by $2$ .

$\frac{16 v}{9} - 16 = 0$

Step 4

Add $16$ to both sides of the equation.

$\frac{16 v}{9} = 16$

Step 5

Multiply both sides of the equation by $\frac{9}{16}$ .

$\frac{9}{16} \cdot \frac{16 v}{9} = \frac{9}{16} \cdot 16$

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 $\frac{9}{16} \cdot \frac{16 v}{9}$ .

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Step 6.1.1.1

Cancel the common factor of $9$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{9}{16} \cdot \frac{16 v}{9} = \frac{9}{16} \cdot 16$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{16} (16 v) = \frac{9}{16} \cdot 16$

Step 6.1.1.2

Cancel the common factor of $16$ .

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Step 6.1.1.2.1

Factor $16$ out of $16 v$ .

$\frac{1}{16} (16 (v)) = \frac{9}{16} \cdot 16$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{16} (16 v) = \frac{9}{16} \cdot 16$

Step 6.1.1.2.3

Rewrite the expression.

$v = \frac{9}{16} \cdot 16$

Step 6.2

Simplify the right side.

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Step 6.2.1

Cancel the common factor of $16$ .

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Step 6.2.1.1

Cancel the common factor.

$v = \frac{9}{16} \cdot 16$

Step 6.2.1.2

Rewrite the expression.

$v = 9$