Algebra Examples

$\frac{x}{3} + \frac{x - 1}{4} = 2 + x$

Step 1

Simplify $\frac{x}{3} + \frac{x - 1}{4}$ .

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Step 1.1

To write $\frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x}{3} \cdot \frac{4}{4} + \frac{x - 1}{4} = 2 + x$

Step 1.2

To write $\frac{x - 1}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{3} \cdot \frac{4}{4} + \frac{x - 1}{4} \cdot \frac{3}{3} = 2 + x$

Step 1.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{x}{3}$ by $\frac{4}{4}$ .

$\frac{x \cdot 4}{3 \cdot 4} + \frac{x - 1}{4} \cdot \frac{3}{3} = 2 + x$

Step 1.3.2

Multiply $3$ by $4$ .

$\frac{x \cdot 4}{12} + \frac{x - 1}{4} \cdot \frac{3}{3} = 2 + x$

Step 1.3.3

Multiply $\frac{x - 1}{4}$ by $\frac{3}{3}$ .

$\frac{x \cdot 4}{12} + \frac{(x - 1) \cdot 3}{4 \cdot 3} = 2 + x$

Step 1.3.4

Multiply $4$ by $3$ .

$\frac{x \cdot 4}{12} + \frac{(x - 1) \cdot 3}{12} = 2 + x$

Step 1.4

Combine the numerators over the common denominator.

$\frac{x \cdot 4 + (x - 1) \cdot 3}{12} = 2 + x$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Move $4$ to the left of $x$ .

$\frac{4 \cdot x + (x - 1) \cdot 3}{12} = 2 + x$

Step 1.5.2

Apply the distributive property.

$\frac{4 x + x \cdot 3 - 1 \cdot 3}{12} = 2 + x$

Step 1.5.3

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

$\frac{4 x + 3 \cdot x - 1 \cdot 3}{12} = 2 + x$

Step 1.5.4

Multiply $- 1$ by $3$ .

$\frac{4 x + 3 \cdot x - 3}{12} = 2 + x$

Step 1.5.5

Add $4 x$ and $3 x$ .

$\frac{7 x - 3}{12} = 2 + x$

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.

$\frac{7 x - 3}{12} - x = 2$

Step 2.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{7 x - 3}{12} - x \cdot \frac{12}{12} = 2$

Step 2.3

Combine $- x$ and $\frac{12}{12}$ .

$\frac{7 x - 3}{12} + \frac{- x \cdot 12}{12} = 2$

Step 2.4

Combine the numerators over the common denominator.

$\frac{7 x - 3 - x \cdot 12}{12} = 2$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $12$ by $- 1$ .

$\frac{7 x - 3 - 12 x}{12} = 2$

Step 2.5.2

Subtract $12 x$ from $7 x$ .

$\frac{- 5 x - 3}{12} = 2$

Step 2.6

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

$\frac{- (5 x) - 3}{12} = 2$

Step 2.7

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{- (5 x) - 1 (3)}{12} = 2$

Step 2.8

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

$\frac{- (5 x + 3)}{12} = 2$

Step 2.9

Rewrite $- (5 x + 3)$ as $- 1 (5 x + 3)$ .

$\frac{- 1 (5 x + 3)}{12} = 2$

Step 2.10

Move the negative in front of the fraction.

$- \frac{5 x + 3}{12} = 2$

Step 3

Multiply both sides of the equation by $- 12$ .

$- 12 (- \frac{5 x + 3}{12}) = - 12 \cdot 2$

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 $- 12 (- \frac{5 x + 3}{12})$ .

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Step 4.1.1.1

Cancel the common factor of $12$ .

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Step 4.1.1.1.1

Move the leading negative in $- \frac{5 x + 3}{12}$ into the numerator.

$- 12 \frac{- (5 x + 3)}{12} = - 12 \cdot 2$

Step 4.1.1.1.2

Factor $12$ out of $- 12$ .

$12 (- 1) \frac{- (5 x + 3)}{12} = - 12 \cdot 2$

Step 4.1.1.1.3

Cancel the common factor.

$12 \cdot - 1 \frac{- (5 x + 3)}{12} = - 12 \cdot 2$

Step 4.1.1.1.4

Rewrite the expression.

$- - (5 x + 3) = - 12 \cdot 2$

Step 4.1.1.2

Multiply.

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Step 4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (5 x + 3) = - 12 \cdot 2$

Step 4.1.1.2.2

Multiply $5 x + 3$ by $1$ .

$5 x + 3 = - 12 \cdot 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- 12$ by $2$ .

$5 x + 3 = - 24$

Step 5

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

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Step 5.1

Subtract $3$ from both sides of the equation.

$5 x = - 24 - 3$

Step 5.2

Subtract $3$ from $- 24$ .

$5 x = - 27$

Step 6

Divide each term in $5 x = - 27$ by $5$ and simplify.

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Step 6.1

Divide each term in $5 x = - 27$ by $5$ .

$\frac{5 x}{5} = \frac{- 27}{5}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $5$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 27}{5}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 27}{5}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

$x = - \frac{27}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{27}{5}$

Decimal Form:

$x = - 5.4$

Mixed Number Form:

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