Precalculus Examples

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

Step 1

Multiply both sides by $5$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $5$ .

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Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

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

Step 2.2

Simplify the right side.

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Step 2.2.1

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

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Step 2.2.1.1

Simplify terms.

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Step 2.2.1.1.1

Apply the distributive property.

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

Step 2.2.1.1.2

Combine $\frac{x}{3}$ and $5$ .

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

Step 2.2.1.1.3

Cancel the common factor of $5$ .

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Step 2.2.1.1.3.1

Cancel the common factor.

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

Step 2.2.1.1.3.2

Rewrite the expression.

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

Step 2.2.1.2

Move $5$ to the left of $x$ .

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

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 $\frac{5 x}{3}$ from both sides of the equation.

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

Step 3.1.2

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

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

Step 3.1.3

Combine $x$ and $\frac{3}{3}$ .

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

Step 3.1.4

Combine the numerators over the common denominator.

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

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Move $3$ to the left of $x$ .

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

Step 3.1.5.2

Subtract $5 x$ from $3 x$ .

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

Step 3.1.6

Move the negative in front of the fraction.

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

Step 3.2

Multiply both sides of the equation by $- \frac{3}{2}$ .

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

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $- \frac{3}{2} (- \frac{2 x}{3})$ .

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Step 3.3.1.1.1

Cancel the common factor of $3$ .

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Step 3.3.1.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Step 3.3.1.1.1.2

Move the leading negative in $- \frac{2 x}{3}$ into the numerator.

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

Step 3.3.1.1.1.3

Factor $3$ out of $- 3$ .

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

Step 3.3.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{2} \cdot \frac{- 2 x}{3} = - \frac{3}{2} \cdot 3$

Step 3.3.1.1.1.5

Rewrite the expression.

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

Step 3.3.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.2.1

Factor $2$ out of $- 2 x$ .

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

Step 3.3.1.1.2.2

Cancel the common factor.

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

Step 3.3.1.1.2.3

Rewrite the expression.

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

Step 3.3.1.1.3

Multiply.

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Step 3.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{3}{2} \cdot 3$

Step 3.3.1.1.3.2

Multiply $x$ by $1$ .

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \frac{3}{2} \cdot 3$ .

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Step 3.3.2.1.1

Multiply $- \frac{3}{2} \cdot 3$ .

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Step 3.3.2.1.1.1

Multiply $3$ by $- 1$ .

$x = - 3 (\frac{3}{2})$

Step 3.3.2.1.1.2

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

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

Step 3.3.2.1.1.3

Multiply $- 3$ by $3$ .

$x = \frac{- 9}{2}$

Step 3.3.2.1.2

Move the negative in front of the fraction.

$x = - \frac{9}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{9}{2}$

Decimal Form:

$x = - 4.5$

Mixed Number Form:

$x = - 4 \frac{1}{2}$