Pre-Algebra Examples

$6 = \frac{5 u + 3}{6} + \frac{2 u + 9}{5}$

Step 1

Rewrite the equation as $\frac{5 u + 3}{6} + \frac{2 u + 9}{5} = 6$ .

$\frac{5 u + 3}{6} + \frac{2 u + 9}{5} = 6$

Step 2

Simplify $\frac{5 u + 3}{6} + \frac{2 u + 9}{5}$ .

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Step 2.1

To write $\frac{5 u + 3}{6}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{5 u + 3}{6} \cdot \frac{5}{5} + \frac{2 u + 9}{5} = 6$

Step 2.2

To write $\frac{2 u + 9}{5}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{5 u + 3}{6} \cdot \frac{5}{5} + \frac{2 u + 9}{5} \cdot \frac{6}{6} = 6$

Step 2.3

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{5 u + 3}{6}$ by $\frac{5}{5}$ .

$\frac{(5 u + 3) \cdot 5}{6 \cdot 5} + \frac{2 u + 9}{5} \cdot \frac{6}{6} = 6$

Step 2.3.2

Multiply $6$ by $5$ .

$\frac{(5 u + 3) \cdot 5}{30} + \frac{2 u + 9}{5} \cdot \frac{6}{6} = 6$

Step 2.3.3

Multiply $\frac{2 u + 9}{5}$ by $\frac{6}{6}$ .

$\frac{(5 u + 3) \cdot 5}{30} + \frac{(2 u + 9) \cdot 6}{5 \cdot 6} = 6$

Step 2.3.4

Multiply $5$ by $6$ .

$\frac{(5 u + 3) \cdot 5}{30} + \frac{(2 u + 9) \cdot 6}{30} = 6$

Step 2.4

Combine the numerators over the common denominator.

$\frac{(5 u + 3) \cdot 5 + (2 u + 9) \cdot 6}{30} = 6$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{5 u \cdot 5 + 3 \cdot 5 + (2 u + 9) \cdot 6}{30} = 6$

Step 2.5.2

Multiply $5$ by $5$ .

$\frac{25 u + 3 \cdot 5 + (2 u + 9) \cdot 6}{30} = 6$

Step 2.5.3

Multiply $3$ by $5$ .

$\frac{25 u + 15 + (2 u + 9) \cdot 6}{30} = 6$

Step 2.5.4

Apply the distributive property.

$\frac{25 u + 15 + 2 u \cdot 6 + 9 \cdot 6}{30} = 6$

Step 2.5.5

Multiply $6$ by $2$ .

$\frac{25 u + 15 + 12 u + 9 \cdot 6}{30} = 6$

Step 2.5.6

Multiply $9$ by $6$ .

$\frac{25 u + 15 + 12 u + 54}{30} = 6$

Step 2.5.7

Add $25 u$ and $12 u$ .

$\frac{37 u + 15 + 54}{30} = 6$

Step 2.5.8

Add $15$ and $54$ .

$\frac{37 u + 69}{30} = 6$

Step 3

Multiply both sides by $30$ .

$\frac{37 u + 69}{30} \cdot 30 = 6 \cdot 30$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $30$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{37 u + 69}{30} \cdot 30 = 6 \cdot 30$

Step 4.1.1.2

Rewrite the expression.

$37 u + 69 = 6 \cdot 30$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $6$ by $30$ .

$37 u + 69 = 180$

Step 5

Solve for $u$ .

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Step 5.1

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

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Step 5.1.1

Subtract $69$ from both sides of the equation.

$37 u = 180 - 69$

Step 5.1.2

Subtract $69$ from $180$ .

$37 u = 111$

Step 5.2

Divide each term in $37 u = 111$ by $37$ and simplify.

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Step 5.2.1

Divide each term in $37 u = 111$ by $37$ .

$\frac{37 u}{37} = \frac{111}{37}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $37$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{37 u}{37} = \frac{111}{37}$

Step 5.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{111}{37}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $111$ by $37$ .

$u = 3$