Precalculus Examples

$\frac{1}{7} \cdot (x - 1) + \frac{6}{14} = 2 + \frac{2}{7} \cdot (3 x - 1)$

Step 1

Simplify $\frac{1}{7} \cdot (x - 1) + \frac{6}{14}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{1}{7} x + \frac{1}{7} \cdot - 1 + \frac{6}{14} = 2 + \frac{2}{7} \cdot (3 x - 1)$

Step 1.1.2

Combine $\frac{1}{7}$ and $x$ .

$\frac{x}{7} + \frac{1}{7} \cdot - 1 + \frac{6}{14} = 2 + \frac{2}{7} \cdot (3 x - 1)$

Step 1.1.3

Combine $\frac{1}{7}$ and $- 1$ .

$\frac{x}{7} + \frac{- 1}{7} + \frac{6}{14} = 2 + \frac{2}{7} \cdot (3 x - 1)$

Step 1.1.4

Move the negative in front of the fraction.

$\frac{x}{7} - \frac{1}{7} + \frac{6}{14} = 2 + \frac{2}{7} \cdot (3 x - 1)$

Step 1.1.5

Cancel the common factor of $6$ and $14$ .

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Step 1.1.5.1

Factor $2$ out of $6$ .

$\frac{x}{7} - \frac{1}{7} + \frac{2 (3)}{14} = 2 + \frac{2}{7} \cdot (3 x - 1)$

Step 1.1.5.2

Cancel the common factors.

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Step 1.1.5.2.1

Factor $2$ out of $14$ .

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

Step 1.1.5.2.2

Cancel the common factor.

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

Step 1.1.5.2.3

Rewrite the expression.

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

Step 1.2

Combine fractions.

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Step 1.2.1

Combine the numerators over the common denominator.

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

Step 1.2.2

Add $- 1$ and $3$ .

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

Step 2

Simplify $2 + \frac{2}{7} \cdot (3 x - 1)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

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

Step 2.1.2

Multiply $\frac{2}{7} (3 x)$ .

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Step 2.1.2.1

Combine $3$ and $\frac{2}{7}$ .

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

Step 2.1.2.2

Multiply $3$ by $2$ .

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

Step 2.1.2.3

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

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

Step 2.1.3

Multiply $\frac{2}{7} \cdot - 1$ .

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Step 2.1.3.1

Combine $\frac{2}{7}$ and $- 1$ .

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

Step 2.1.3.2

Multiply $2$ by $- 1$ .

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

Step 2.1.4

Move the negative in front of the fraction.

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

Step 2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $2$ by $7$ .

$\frac{x + 2}{7} = \frac{6 x}{7} + \frac{14 - 2}{7}$

Step 2.5.2

Subtract $2$ from $14$ .

$\frac{x + 2}{7} = \frac{6 x}{7} + \frac{12}{7}$

Step 3

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

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Step 3.1

Subtract $\frac{6 x}{7}$ from both sides of the equation.

$\frac{x + 2}{7} - \frac{6 x}{7} = \frac{12}{7}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{x + 2 - 6 x}{7} = \frac{12}{7}$

Step 3.3

Subtract $6 x$ from $x$ .

$\frac{- 5 x + 2}{7} = \frac{12}{7}$

Step 3.4

Factor $- 1$ out of $- 5 x$ .

$\frac{- (5 x) + 2}{7} = \frac{12}{7}$

Step 3.5

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{- (5 x) - 1 (- 2)}{7} = \frac{12}{7}$

Step 3.6

Factor $- 1$ out of $- (5 x) - 1 (- 2)$ .

$\frac{- (5 x - 2)}{7} = \frac{12}{7}$

Step 3.7

Rewrite $- (5 x - 2)$ as $- 1 (5 x - 2)$ .

$\frac{- 1 (5 x - 2)}{7} = \frac{12}{7}$

Step 3.8

Move the negative in front of the fraction.

$- \frac{5 x - 2}{7} = \frac{12}{7}$

Step 4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- (5 x - 2) = 12$

Step 5

Simplify $- (5 x - 2)$ .

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Step 5.1

Apply the distributive property.

$- (5 x) - - 2 = 12$

Step 5.2

Multiply.

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Step 5.2.1

Multiply $5$ by $- 1$ .

$- 5 x - - 2 = 12$

Step 5.2.2

Multiply $- 1$ by $- 2$ .

$- 5 x + 2 = 12$

Step 6

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

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Step 6.1

Subtract $2$ from both sides of the equation.

$- 5 x = 12 - 2$

Step 6.2

Subtract $2$ from $12$ .

$- 5 x = 10$

Step 7

Divide each term in $- 5 x = 10$ by $- 5$ and simplify.

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Step 7.1

Divide each term in $- 5 x = 10$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{10}{- 5}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 5$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{10}{- 5}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{- 5}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $10$ by $- 5$ .

$x = - 2$