Precalculus Examples

$\frac{1}{2} x - \frac{3}{7} = 1 - \frac{x}{3}$

Step 1

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

$\frac{x}{2} - \frac{3}{7} = 1 - \frac{x}{3}$

Step 2

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

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Step 2.1

Add $\frac{x}{3}$ to both sides of the equation.

$\frac{x}{2} - \frac{3}{7} + \frac{x}{3} = 1$

Step 2.2

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

$\frac{x}{2} \cdot \frac{3}{3} + \frac{x}{3} - \frac{3}{7} = 1$

Step 2.3

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

$\frac{x}{2} \cdot \frac{3}{3} + \frac{x}{3} \cdot \frac{2}{2} - \frac{3}{7} = 1$

Step 2.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

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

$\frac{x \cdot 3}{2 \cdot 3} + \frac{x}{3} \cdot \frac{2}{2} - \frac{3}{7} = 1$

Step 2.4.2

Multiply $2$ by $3$ .

$\frac{x \cdot 3}{6} + \frac{x}{3} \cdot \frac{2}{2} - \frac{3}{7} = 1$

Step 2.4.3

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

$\frac{x \cdot 3}{6} + \frac{x \cdot 2}{3 \cdot 2} - \frac{3}{7} = 1$

Step 2.4.4

Multiply $3$ by $2$ .

$\frac{x \cdot 3}{6} + \frac{x \cdot 2}{6} - \frac{3}{7} = 1$

Step 2.5

Combine the numerators over the common denominator.

$\frac{x \cdot 3 + x \cdot 2}{6} - \frac{3}{7} = 1$

Step 2.6

Reorder $x$ and $2$ .

$\frac{3 \cdot x + 2 \cdot x}{6} - \frac{3}{7} = 1$

Step 2.7

Add $3 \cdot x$ and $2 \cdot x$ .

$\frac{5 x}{6} - \frac{3}{7} = 1$

Step 3

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

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Step 3.1

Add $\frac{3}{7}$ to both sides of the equation.

$\frac{5 x}{6} = 1 + \frac{3}{7}$

Step 3.2

Write $1$ as a fraction with a common denominator.

$\frac{5 x}{6} = \frac{7}{7} + \frac{3}{7}$

Step 3.3

Combine the numerators over the common denominator.

$\frac{5 x}{6} = \frac{7 + 3}{7}$

Step 3.4

Add $7$ and $3$ .

$\frac{5 x}{6} = \frac{10}{7}$

Step 4

Multiply both sides of the equation by $\frac{6}{5}$ .

$\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} \cdot \frac{10}{7}$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $\frac{6}{5} \cdot \frac{5 x}{6}$ .

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Step 5.1.1.1

Cancel the common factor of $6$ .

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Step 5.1.1.1.1

Cancel the common factor.

$\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} \cdot \frac{10}{7}$

Step 5.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{6}{5} \cdot \frac{10}{7}$

Step 5.1.1.2

Cancel the common factor of $5$ .

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Step 5.1.1.2.1

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) = \frac{6}{5} \cdot \frac{10}{7}$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{6}{5} \cdot \frac{10}{7}$

Step 5.1.1.2.3

Rewrite the expression.

$x = \frac{6}{5} \cdot \frac{10}{7}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{6}{5} \cdot \frac{10}{7}$ .

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Step 5.2.1.1

Cancel the common factor of $5$ .

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Step 5.2.1.1.1

Factor $5$ out of $10$ .

$x = \frac{6}{5} \cdot \frac{5 (2)}{7}$

Step 5.2.1.1.2

Cancel the common factor.

$x = \frac{6}{5} \cdot \frac{5 \cdot 2}{7}$

Step 5.2.1.1.3

Rewrite the expression.

$x = 6 (\frac{2}{7})$

Step 5.2.1.2

Combine $6$ and $\frac{2}{7}$ .

$x = \frac{6 \cdot 2}{7}$

Step 5.2.1.3

Multiply $6$ by $2$ .

$x = \frac{12}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{12}{7}$

Decimal Form:

$x = 1.\overline{714285}$

Mixed Number Form:

$x = 1 \frac{5}{7}$