Algebra Examples

$\frac{2 x - 1}{6} - \frac{x + 5}{18} = x + 1$

Step 1

Simplify $\frac{2 x - 1}{6} - \frac{x + 5}{18}$ .

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Step 1.1

To write $\frac{2 x - 1}{6}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 x - 1}{6} \cdot \frac{3}{3} - \frac{x + 5}{18} = x + 1$

Step 1.2

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1

Multiply $\frac{2 x - 1}{6}$ by $\frac{3}{3}$ .

$\frac{(2 x - 1) \cdot 3}{6 \cdot 3} - \frac{x + 5}{18} = x + 1$

Step 1.2.2

Multiply $6$ by $3$ .

$\frac{(2 x - 1) \cdot 3}{18} - \frac{x + 5}{18} = x + 1$

Step 1.3

Combine the numerators over the common denominator.

$\frac{(2 x - 1) \cdot 3 - (x + 5)}{18} = x + 1$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Apply the distributive property.

$\frac{2 x \cdot 3 - 1 \cdot 3 - (x + 5)}{18} = x + 1$

Step 1.4.2

Multiply $3$ by $2$ .

$\frac{6 x - 1 \cdot 3 - (x + 5)}{18} = x + 1$

Step 1.4.3

Multiply $- 1$ by $3$ .

$\frac{6 x - 3 - (x + 5)}{18} = x + 1$

Step 1.4.4

Apply the distributive property.

$\frac{6 x - 3 - x - 1 \cdot 5}{18} = x + 1$

Step 1.4.5

Multiply $- 1$ by $5$ .

$\frac{6 x - 3 - x - 5}{18} = x + 1$

Step 1.4.6

Subtract $x$ from $6 x$ .

$\frac{5 x - 3 - 5}{18} = x + 1$

Step 1.4.7

Subtract $5$ from $- 3$ .

$\frac{5 x - 8}{18} = x + 1$

Step 2

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

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Step 2.1

Subtract $x$ from both sides of the equation.

$\frac{5 x - 8}{18} - x = 1$

Step 2.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{18}{18}$ .

$\frac{5 x - 8}{18} - x \cdot \frac{18}{18} = 1$

Step 2.3

Combine $- x$ and $\frac{18}{18}$ .

$\frac{5 x - 8}{18} + \frac{- x \cdot 18}{18} = 1$

Step 2.4

Combine the numerators over the common denominator.

$\frac{5 x - 8 - x \cdot 18}{18} = 1$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $18$ by $- 1$ .

$\frac{5 x - 8 - 18 x}{18} = 1$

Step 2.5.2

Subtract $18 x$ from $5 x$ .

$\frac{- 13 x - 8}{18} = 1$

Step 2.6

Factor $- 1$ out of $- 13 x$ .

$\frac{- (13 x) - 8}{18} = 1$

Step 2.7

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{- (13 x) - 1 (8)}{18} = 1$

Step 2.8

Factor $- 1$ out of $- (13 x) - 1 (8)$ .

$\frac{- (13 x + 8)}{18} = 1$

Step 2.9

Rewrite $- (13 x + 8)$ as $- 1 (13 x + 8)$ .

$\frac{- 1 (13 x + 8)}{18} = 1$

Step 2.10

Move the negative in front of the fraction.

$- \frac{13 x + 8}{18} = 1$

Step 3

Multiply both sides of the equation by $- 18$ .

$- 18 (- \frac{13 x + 8}{18}) = - 18 \cdot 1$

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 $- 18 (- \frac{13 x + 8}{18})$ .

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Step 4.1.1.1

Cancel the common factor of $18$ .

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Step 4.1.1.1.1

Move the leading negative in $- \frac{13 x + 8}{18}$ into the numerator.

$- 18 \frac{- (13 x + 8)}{18} = - 18 \cdot 1$

Step 4.1.1.1.2

Factor $18$ out of $- 18$ .

$18 (- 1) \frac{- (13 x + 8)}{18} = - 18 \cdot 1$

Step 4.1.1.1.3

Cancel the common factor.

$18 \cdot - 1 \frac{- (13 x + 8)}{18} = - 18 \cdot 1$

Step 4.1.1.1.4

Rewrite the expression.

$- - (13 x + 8) = - 18 \cdot 1$

Step 4.1.1.2

Multiply.

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Step 4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (13 x + 8) = - 18 \cdot 1$

Step 4.1.1.2.2

Multiply $13 x + 8$ by $1$ .

$13 x + 8 = - 18 \cdot 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- 18$ by $1$ .

$13 x + 8 = - 18$

Step 5

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

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Step 5.1

Subtract $8$ from both sides of the equation.

$13 x = - 18 - 8$

Step 5.2

Subtract $8$ from $- 18$ .

$13 x = - 26$

Step 6

Divide each term in $13 x = - 26$ by $13$ and simplify.

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Step 6.1

Divide each term in $13 x = - 26$ by $13$ .

$\frac{13 x}{13} = \frac{- 26}{13}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $13$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{13 x}{13} = \frac{- 26}{13}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 26}{13}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $- 26$ by $13$ .

$x = - 2$