Pre-Algebra Examples

$\frac{x}{3} - 5 = \frac{x}{5} - 3$

Step 1

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

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Step 1.1

Subtract $\frac{x}{5}$ from both sides of the equation.

$\frac{x}{3} - 5 - \frac{x}{5} = - 3$

Step 1.2

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

$\frac{x}{3} \cdot \frac{5}{5} - \frac{x}{5} - 5 = - 3$

Step 1.3

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

$\frac{x}{3} \cdot \frac{5}{5} - \frac{x}{5} \cdot \frac{3}{3} - 5 = - 3$

Step 1.4

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

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Step 1.4.1

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

$\frac{x \cdot 5}{3 \cdot 5} - \frac{x}{5} \cdot \frac{3}{3} - 5 = - 3$

Step 1.4.2

Multiply $3$ by $5$ .

$\frac{x \cdot 5}{15} - \frac{x}{5} \cdot \frac{3}{3} - 5 = - 3$

Step 1.4.3

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

$\frac{x \cdot 5}{15} - \frac{x \cdot 3}{5 \cdot 3} - 5 = - 3$

Step 1.4.4

Multiply $5$ by $3$ .

$\frac{x \cdot 5}{15} - \frac{x \cdot 3}{15} - 5 = - 3$

Step 1.5

Combine the numerators over the common denominator.

$\frac{x \cdot 5 - x \cdot 3}{15} - 5 = - 3$

Step 1.6

Simplify each term.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Factor $x$ out of $x \cdot 5 - x \cdot 3$ .

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Step 1.6.1.1.1

Factor $x$ out of $x \cdot 5$ .

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

Step 1.6.1.1.2

Factor $x$ out of $- x \cdot 3$ .

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

Step 1.6.1.1.3

Factor $x$ out of $x (5) + x (- 1 \cdot 3)$ .

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

Step 1.6.1.2

Multiply $- 1$ by $3$ .

$\frac{x (5 - 3)}{15} - 5 = - 3$

Step 1.6.1.3

Subtract $3$ from $5$ .

$\frac{x \cdot 2}{15} - 5 = - 3$

Step 1.6.2

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

$\frac{2 x}{15} - 5 = - 3$

Step 2

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

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Step 2.1

Add $5$ to both sides of the equation.

$\frac{2 x}{15} = - 3 + 5$

Step 2.2

Add $- 3$ and $5$ .

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

Step 3

Multiply both sides of the equation by $\frac{15}{2}$ .

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

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{15}{2} \cdot \frac{2 x}{15}$ .

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Step 4.1.1.1

Cancel the common factor of $15$ .

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Step 4.1.1.1.1

Cancel the common factor.

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

Step 4.1.1.1.2

Rewrite the expression.

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

Step 4.1.1.2

Cancel the common factor of $2$ .

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Step 4.1.1.2.1

Factor $2$ out of $2 x$ .

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

Step 4.1.1.2.2

Cancel the common factor.

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

Step 4.1.1.2.3

Rewrite the expression.

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

Step 4.2

Simplify the right side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Rewrite the expression.

$x = 15$