Algebra Examples

$\frac{x}{5} + \frac{x}{3} - 1 = \frac{x}{2}$

Step 1

Multiply both sides by $2$ .

$(\frac{x}{5} + \frac{x}{3} - 1) \cdot 2 = \frac{x}{2} \cdot 2$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{x}{5} + \frac{x}{3} - 1) \cdot 2$ .

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Step 2.1.1.1

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

$(\frac{x}{5} \cdot \frac{3}{3} + \frac{x}{3} - 1) \cdot 2 = \frac{x}{2} \cdot 2$

Step 2.1.1.2

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

$(\frac{x}{5} \cdot \frac{3}{3} + \frac{x}{3} \cdot \frac{5}{5} - 1) \cdot 2 = \frac{x}{2} \cdot 2$

Step 2.1.1.3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1.1.3.1

Multiply $\frac{x}{5}$ by $\frac{3}{3}$ .

$(\frac{x \cdot 3}{5 \cdot 3} + \frac{x}{3} \cdot \frac{5}{5} - 1) \cdot 2 = \frac{x}{2} \cdot 2$

Step 2.1.1.3.2

Multiply $5$ by $3$ .

$(\frac{x \cdot 3}{15} + \frac{x}{3} \cdot \frac{5}{5} - 1) \cdot 2 = \frac{x}{2} \cdot 2$

Step 2.1.1.3.3

Multiply $\frac{x}{3}$ by $\frac{5}{5}$ .

$(\frac{x \cdot 3}{15} + \frac{x \cdot 5}{3 \cdot 5} - 1) \cdot 2 = \frac{x}{2} \cdot 2$

Step 2.1.1.3.4

Multiply $3$ by $5$ .

$(\frac{x \cdot 3}{15} + \frac{x \cdot 5}{15} - 1) \cdot 2 = \frac{x}{2} \cdot 2$

Step 2.1.1.4

Combine the numerators over the common denominator.

$(\frac{x \cdot 3 + x \cdot 5}{15} - 1) \cdot 2 = \frac{x}{2} \cdot 2$

Step 2.1.1.5

Reorder $x$ and $5$ .

$(\frac{3 \cdot x + 5 \cdot x}{15} - 1) \cdot 2 = \frac{x}{2} \cdot 2$

Step 2.1.1.6

Add $3 \cdot x$ and $5 \cdot x$ .

$(\frac{8 \cdot x}{15} - 1) \cdot 2 = \frac{x}{2} \cdot 2$

Step 2.1.1.7

Apply the distributive property.

$\frac{8 x}{15} \cdot 2 - 1 \cdot 2 = \frac{x}{2} \cdot 2$

Step 2.1.1.8

Multiply $\frac{8 x}{15} \cdot 2$ .

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Step 2.1.1.8.1

Combine $\frac{8 x}{15}$ and $2$ .

$\frac{8 x \cdot 2}{15} - 1 \cdot 2 = \frac{x}{2} \cdot 2$

Step 2.1.1.8.2

Multiply $2$ by $8$ .

$\frac{16 x}{15} - 1 \cdot 2 = \frac{x}{2} \cdot 2$

Step 2.1.1.9

Multiply $- 1$ by $2$ .

$\frac{16 x}{15} - 2 = \frac{x}{2} \cdot 2$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{16 x}{15} - 2 = \frac{x}{2} \cdot 2$

Step 2.2.1.2

Rewrite the expression.

$\frac{16 x}{15} - 2 = x$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $x$ from both sides of the equation.

$\frac{16 x}{15} - 2 - x = 0$

Step 3.1.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{16 x}{15} - x \cdot \frac{15}{15} - 2 = 0$

Step 3.1.3

Combine $- x$ and $\frac{15}{15}$ .

$\frac{16 x}{15} + \frac{- x \cdot 15}{15} - 2 = 0$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{16 x - x \cdot 15}{15} - 2 = 0$

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

Simplify the numerator.

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Step 3.1.5.1.1

Factor $x$ out of $16 x - x \cdot 15$ .

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Step 3.1.5.1.1.1

Factor $x$ out of $16 x$ .

$\frac{x \cdot 16 - x \cdot 15}{15} - 2 = 0$

Step 3.1.5.1.1.2

Factor $x$ out of $- x \cdot 15$ .

$\frac{x \cdot 16 + x (- 1 \cdot 15)}{15} - 2 = 0$

Step 3.1.5.1.1.3

Factor $x$ out of $x \cdot 16 + x (- 1 \cdot 15)$ .

$\frac{x (16 - 1 \cdot 15)}{15} - 2 = 0$

Step 3.1.5.1.2

Multiply $- 1$ by $15$ .

$\frac{x (16 - 15)}{15} - 2 = 0$

Step 3.1.5.1.3

Subtract $15$ from $16$ .

$\frac{x \cdot 1}{15} - 2 = 0$

Step 3.1.5.2

Multiply $x$ by $1$ .

$\frac{x}{15} - 2 = 0$

Step 3.2

Add $2$ to both sides of the equation.

$\frac{x}{15} = 2$

Step 3.3

Multiply both sides of the equation by $15$ .

$15 \frac{x}{15} = 15 \cdot 2$

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Cancel the common factor of $15$ .

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Step 3.4.1.1.1

Cancel the common factor.

$15 \frac{x}{15} = 15 \cdot 2$

Step 3.4.1.1.2

Rewrite the expression.

$x = 15 \cdot 2$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Multiply $15$ by $2$ .

$x = 30$