Algebra Examples

$\frac{x - 5}{2} + \frac{x - 1}{8} = \frac{x - 3}{4} + \frac{x - 4}{3}$

Step 1

Simplify $\frac{x - 5}{2} + \frac{x - 1}{8}$ .

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Step 1.1

To write $\frac{x - 5}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x - 5}{2} \cdot \frac{4}{4} + \frac{x - 1}{8} = \frac{x - 3}{4} + \frac{x - 4}{3}$

Step 1.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1

Multiply $\frac{x - 5}{2}$ by $\frac{4}{4}$ .

$\frac{(x - 5) \cdot 4}{2 \cdot 4} + \frac{x - 1}{8} = \frac{x - 3}{4} + \frac{x - 4}{3}$

Step 1.2.2

Multiply $2$ by $4$ .

$\frac{(x - 5) \cdot 4}{8} + \frac{x - 1}{8} = \frac{x - 3}{4} + \frac{x - 4}{3}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{(x - 5) \cdot 4 + x - 1}{8} = \frac{x - 3}{4} + \frac{x - 4}{3}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Apply the distributive property.

$\frac{x \cdot 4 - 5 \cdot 4 + x - 1}{8} = \frac{x - 3}{4} + \frac{x - 4}{3}$

Step 1.4.2

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

$\frac{4 \cdot x - 5 \cdot 4 + x - 1}{8} = \frac{x - 3}{4} + \frac{x - 4}{3}$

Step 1.4.3

Multiply $- 5$ by $4$ .

$\frac{4 \cdot x - 20 + x - 1}{8} = \frac{x - 3}{4} + \frac{x - 4}{3}$

Step 1.4.4

Add $4 x$ and $x$ .

$\frac{5 x - 20 - 1}{8} = \frac{x - 3}{4} + \frac{x - 4}{3}$

Step 1.4.5

Subtract $1$ from $- 20$ .

$\frac{5 x - 21}{8} = \frac{x - 3}{4} + \frac{x - 4}{3}$

Step 2

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

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Step 2.1

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

$\frac{5 x - 21}{8} = \frac{x - 3}{4} \cdot \frac{3}{3} + \frac{x - 4}{3}$

Step 2.2

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

$\frac{5 x - 21}{8} = \frac{x - 3}{4} \cdot \frac{3}{3} + \frac{x - 4}{3} \cdot \frac{4}{4}$

Step 2.3

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

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Step 2.3.1

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

$\frac{5 x - 21}{8} = \frac{(x - 3) \cdot 3}{4 \cdot 3} + \frac{x - 4}{3} \cdot \frac{4}{4}$

Step 2.3.2

Multiply $4$ by $3$ .

$\frac{5 x - 21}{8} = \frac{(x - 3) \cdot 3}{12} + \frac{x - 4}{3} \cdot \frac{4}{4}$

Step 2.3.3

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

$\frac{5 x - 21}{8} = \frac{(x - 3) \cdot 3}{12} + \frac{(x - 4) \cdot 4}{3 \cdot 4}$

Step 2.3.4

Multiply $3$ by $4$ .

$\frac{5 x - 21}{8} = \frac{(x - 3) \cdot 3}{12} + \frac{(x - 4) \cdot 4}{12}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{5 x - 21}{8} = \frac{(x - 3) \cdot 3 + (x - 4) \cdot 4}{12}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{5 x - 21}{8} = \frac{x \cdot 3 - 3 \cdot 3 + (x - 4) \cdot 4}{12}$

Step 2.5.2

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

$\frac{5 x - 21}{8} = \frac{3 \cdot x - 3 \cdot 3 + (x - 4) \cdot 4}{12}$

Step 2.5.3

Multiply $- 3$ by $3$ .

$\frac{5 x - 21}{8} = \frac{3 \cdot x - 9 + (x - 4) \cdot 4}{12}$

Step 2.5.4

Apply the distributive property.

$\frac{5 x - 21}{8} = \frac{3 x - 9 + x \cdot 4 - 4 \cdot 4}{12}$

Step 2.5.5

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

$\frac{5 x - 21}{8} = \frac{3 x - 9 + 4 \cdot x - 4 \cdot 4}{12}$

Step 2.5.6

Multiply $- 4$ by $4$ .

$\frac{5 x - 21}{8} = \frac{3 x - 9 + 4 \cdot x - 16}{12}$

Step 2.5.7

Add $3 x$ and $4 x$ .

$\frac{5 x - 21}{8} = \frac{7 x - 9 - 16}{12}$

Step 2.5.8

Subtract $16$ from $- 9$ .

$\frac{5 x - 21}{8} = \frac{7 x - 25}{12}$

Step 3

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

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Step 3.1

Subtract $\frac{7 x - 25}{12}$ from both sides of the equation.

$\frac{5 x - 21}{8} - \frac{7 x - 25}{12} = 0$

Step 3.2

To write $\frac{5 x - 21}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5 x - 21}{8} \cdot \frac{3}{3} - \frac{7 x - 25}{12} = 0$

Step 3.3

To write $- \frac{7 x - 25}{12}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5 x - 21}{8} \cdot \frac{3}{3} - \frac{7 x - 25}{12} \cdot \frac{2}{2} = 0$

Step 3.4

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{5 x - 21}{8}$ by $\frac{3}{3}$ .

$\frac{(5 x - 21) \cdot 3}{8 \cdot 3} - \frac{7 x - 25}{12} \cdot \frac{2}{2} = 0$

Step 3.4.2

Multiply $8$ by $3$ .

$\frac{(5 x - 21) \cdot 3}{24} - \frac{7 x - 25}{12} \cdot \frac{2}{2} = 0$

Step 3.4.3

Multiply $\frac{7 x - 25}{12}$ by $\frac{2}{2}$ .

$\frac{(5 x - 21) \cdot 3}{24} - \frac{(7 x - 25) \cdot 2}{12 \cdot 2} = 0$

Step 3.4.4

Multiply $12$ by $2$ .

$\frac{(5 x - 21) \cdot 3}{24} - \frac{(7 x - 25) \cdot 2}{24} = 0$

Step 3.5

Combine the numerators over the common denominator.

$\frac{(5 x - 21) \cdot 3 - (7 x - 25) \cdot 2}{24} = 0$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

$\frac{5 x \cdot 3 - 21 \cdot 3 - (7 x - 25) \cdot 2}{24} = 0$

Step 3.6.2

Multiply $3$ by $5$ .

$\frac{15 x - 21 \cdot 3 - (7 x - 25) \cdot 2}{24} = 0$

Step 3.6.3

Multiply $- 21$ by $3$ .

$\frac{15 x - 63 - (7 x - 25) \cdot 2}{24} = 0$

Step 3.6.4

Apply the distributive property.

$\frac{15 x - 63 + (- (7 x) - - 25) \cdot 2}{24} = 0$

Step 3.6.5

Multiply $7$ by $- 1$ .

$\frac{15 x - 63 + (- 7 x - - 25) \cdot 2}{24} = 0$

Step 3.6.6

Multiply $- 1$ by $- 25$ .

$\frac{15 x - 63 + (- 7 x + 25) \cdot 2}{24} = 0$

Step 3.6.7

Apply the distributive property.

$\frac{15 x - 63 - 7 x \cdot 2 + 25 \cdot 2}{24} = 0$

Step 3.6.8

Multiply $2$ by $- 7$ .

$\frac{15 x - 63 - 14 x + 25 \cdot 2}{24} = 0$

Step 3.6.9

Multiply $25$ by $2$ .

$\frac{15 x - 63 - 14 x + 50}{24} = 0$

Step 3.6.10

Subtract $14 x$ from $15 x$ .

$\frac{x - 63 + 50}{24} = 0$

Step 3.6.11

Add $- 63$ and $50$ .

$\frac{x - 13}{24} = 0$

Step 4

Set the numerator equal to zero.

$x - 13 = 0$

Step 5

Add $13$ to both sides of the equation.

$x = 13$