Algebra Examples

$\frac{8 x + 7}{6} + \frac{3 - 2 x}{4} - \frac{5 x - 2}{2} = 32$

Step 1

Simplify $\frac{8 x + 7}{6} + \frac{3 - 2 x}{4} - \frac{5 x - 2}{2}$ .

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Step 1.1

Find the common denominator.

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Step 1.1.1

Multiply $\frac{8 x + 7}{6}$ by $\frac{2}{2}$ .

$\frac{8 x + 7}{6} \cdot \frac{2}{2} + \frac{3 - 2 x}{4} - \frac{5 x - 2}{2} = 32$

Step 1.1.2

Multiply $\frac{8 x + 7}{6}$ by $\frac{2}{2}$ .

$\frac{(8 x + 7) \cdot 2}{6 \cdot 2} + \frac{3 - 2 x}{4} - \frac{5 x - 2}{2} = 32$

Step 1.1.3

Multiply $\frac{3 - 2 x}{4}$ by $\frac{3}{3}$ .

$\frac{(8 x + 7) \cdot 2}{6 \cdot 2} + \frac{3 - 2 x}{4} \cdot \frac{3}{3} - \frac{5 x - 2}{2} = 32$

Step 1.1.4

Multiply $\frac{3 - 2 x}{4}$ by $\frac{3}{3}$ .

$\frac{(8 x + 7) \cdot 2}{6 \cdot 2} + \frac{(3 - 2 x) \cdot 3}{4 \cdot 3} - \frac{5 x - 2}{2} = 32$

Step 1.1.5

Multiply $\frac{5 x - 2}{2}$ by $\frac{6}{6}$ .

$\frac{(8 x + 7) \cdot 2}{6 \cdot 2} + \frac{(3 - 2 x) \cdot 3}{4 \cdot 3} - (\frac{5 x - 2}{2} \cdot \frac{6}{6}) = 32$

Step 1.1.6

Multiply $\frac{5 x - 2}{2}$ by $\frac{6}{6}$ .

$\frac{(8 x + 7) \cdot 2}{6 \cdot 2} + \frac{(3 - 2 x) \cdot 3}{4 \cdot 3} - \frac{(5 x - 2) \cdot 6}{2 \cdot 6} = 32$

Step 1.1.7

Reorder the factors of $6 \cdot 2$ .

$\frac{(8 x + 7) \cdot 2}{2 \cdot 6} + \frac{(3 - 2 x) \cdot 3}{4 \cdot 3} - \frac{(5 x - 2) \cdot 6}{2 \cdot 6} = 32$

Step 1.1.8

Multiply $2$ by $6$ .

$\frac{(8 x + 7) \cdot 2}{12} + \frac{(3 - 2 x) \cdot 3}{4 \cdot 3} - \frac{(5 x - 2) \cdot 6}{2 \cdot 6} = 32$

Step 1.1.9

Reorder the factors of $4 \cdot 3$ .

$\frac{(8 x + 7) \cdot 2}{12} + \frac{(3 - 2 x) \cdot 3}{3 \cdot 4} - \frac{(5 x - 2) \cdot 6}{2 \cdot 6} = 32$

Step 1.1.10

Multiply $3$ by $4$ .

$\frac{(8 x + 7) \cdot 2}{12} + \frac{(3 - 2 x) \cdot 3}{12} - \frac{(5 x - 2) \cdot 6}{2 \cdot 6} = 32$

Step 1.1.11

Multiply $2$ by $6$ .

$\frac{(8 x + 7) \cdot 2}{12} + \frac{(3 - 2 x) \cdot 3}{12} - \frac{(5 x - 2) \cdot 6}{12} = 32$

Step 1.2

Combine the numerators over the common denominator.

$\frac{(8 x + 7) \cdot 2 + (3 - 2 x) \cdot 3 - (5 x - 2) \cdot 6}{12} = 32$

Step 1.3

Simplify each term.

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Step 1.3.1

Apply the distributive property.

$\frac{8 x \cdot 2 + 7 \cdot 2 + (3 - 2 x) \cdot 3 - (5 x - 2) \cdot 6}{12} = 32$

Step 1.3.2

Multiply $2$ by $8$ .

$\frac{16 x + 7 \cdot 2 + (3 - 2 x) \cdot 3 - (5 x - 2) \cdot 6}{12} = 32$

Step 1.3.3

Multiply $7$ by $2$ .

$\frac{16 x + 14 + (3 - 2 x) \cdot 3 - (5 x - 2) \cdot 6}{12} = 32$

Step 1.3.4

Apply the distributive property.

$\frac{16 x + 14 + 3 \cdot 3 - 2 x \cdot 3 - (5 x - 2) \cdot 6}{12} = 32$

Step 1.3.5

Multiply $3$ by $3$ .

$\frac{16 x + 14 + 9 - 2 x \cdot 3 - (5 x - 2) \cdot 6}{12} = 32$

Step 1.3.6

Multiply $3$ by $- 2$ .

$\frac{16 x + 14 + 9 - 6 x - (5 x - 2) \cdot 6}{12} = 32$

Step 1.3.7

Apply the distributive property.

$\frac{16 x + 14 + 9 - 6 x + (- (5 x) - - 2) \cdot 6}{12} = 32$

Step 1.3.8

Multiply $5$ by $- 1$ .

$\frac{16 x + 14 + 9 - 6 x + (- 5 x - - 2) \cdot 6}{12} = 32$

Step 1.3.9

Multiply $- 1$ by $- 2$ .

$\frac{16 x + 14 + 9 - 6 x + (- 5 x + 2) \cdot 6}{12} = 32$

Step 1.3.10

Apply the distributive property.

$\frac{16 x + 14 + 9 - 6 x - 5 x \cdot 6 + 2 \cdot 6}{12} = 32$

Step 1.3.11

Multiply $6$ by $- 5$ .

$\frac{16 x + 14 + 9 - 6 x - 30 x + 2 \cdot 6}{12} = 32$

Step 1.3.12

Multiply $2$ by $6$ .

$\frac{16 x + 14 + 9 - 6 x - 30 x + 12}{12} = 32$

Step 1.4

Simplify terms.

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Step 1.4.1

Subtract $6 x$ from $16 x$ .

$\frac{10 x + 14 + 9 - 30 x + 12}{12} = 32$

Step 1.4.2

Subtract $30 x$ from $10 x$ .

$\frac{- 20 x + 14 + 9 + 12}{12} = 32$

Step 1.4.3

Simplify by adding numbers.

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Step 1.4.3.1

Add $14$ and $9$ .

$\frac{- 20 x + 23 + 12}{12} = 32$

Step 1.4.3.2

Add $23$ and $12$ .

$\frac{- 20 x + 35}{12} = 32$

Step 1.4.4

Factor $5$ out of $- 20 x + 35$ .

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Step 1.4.4.1

Factor $5$ out of $- 20 x$ .

$\frac{5 (- 4 x) + 35}{12} = 32$

Step 1.4.4.2

Factor $5$ out of $35$ .

$\frac{5 (- 4 x) + 5 (7)}{12} = 32$

Step 1.4.4.3

Factor $5$ out of $5 (- 4 x) + 5 (7)$ .

$\frac{5 (- 4 x + 7)}{12} = 32$

Step 1.4.5

Factor $- 1$ out of $- 4 x$ .

$\frac{5 (- (4 x) + 7)}{12} = 32$

Step 1.4.6

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{5 (- (4 x) - 1 (- 7))}{12} = 32$

Step 1.4.7

Factor $- 1$ out of $- (4 x) - 1 (- 7)$ .

$\frac{5 (- (4 x - 7))}{12} = 32$

Step 1.4.8

Simplify the expression.

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Step 1.4.8.1

Rewrite $- (4 x - 7)$ as $- 1 (4 x - 7)$ .

$\frac{5 (- 1 (4 x - 7))}{12} = 32$

Step 1.4.8.2

Move the negative in front of the fraction.

$- \frac{5 (4 x - 7)}{12} = 32$

Step 2

Multiply both sides of the equation by $- \frac{12}{5}$ .

$- \frac{12}{5} (- \frac{5 (4 x - 7)}{12}) = - \frac{12}{5} \cdot 32$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $- \frac{12}{5} (- \frac{5 (4 x - 7)}{12})$ .

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Step 3.1.1.1

Cancel the common factor of $12$ .

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Step 3.1.1.1.1

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

$\frac{- 12}{5} (- \frac{5 (4 x - 7)}{12}) = - \frac{12}{5} \cdot 32$

Step 3.1.1.1.2

Move the leading negative in $- \frac{5 (4 x - 7)}{12}$ into the numerator.

$\frac{- 12}{5} \cdot \frac{- 5 (4 x - 7)}{12} = - \frac{12}{5} \cdot 32$

Step 3.1.1.1.3

Factor $12$ out of $- 12$ .

$\frac{12 (- 1)}{5} \cdot \frac{- 5 (4 x - 7)}{12} = - \frac{12}{5} \cdot 32$

Step 3.1.1.1.4

Cancel the common factor.

$\frac{12 \cdot - 1}{5} \cdot \frac{- 5 (4 x - 7)}{12} = - \frac{12}{5} \cdot 32$

Step 3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{5} (- 5 (4 x - 7)) = - \frac{12}{5} \cdot 32$

Step 3.1.1.2

Cancel the common factor of $5$ .

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Step 3.1.1.2.1

Factor $5$ out of $- 5 (4 x - 7)$ .

$\frac{- 1}{5} (5 (- (4 x - 7))) = - \frac{12}{5} \cdot 32$

Step 3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{5} (5 (- (4 x - 7))) = - \frac{12}{5} \cdot 32$

Step 3.1.1.2.3

Rewrite the expression.

$- - (4 x - 7) = - \frac{12}{5} \cdot 32$

Step 3.1.1.3

Multiply.

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Step 3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 (4 x - 7) = - \frac{12}{5} \cdot 32$

Step 3.1.1.3.2

Multiply $4 x - 7$ by $1$ .

$4 x - 7 = - \frac{12}{5} \cdot 32$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- \frac{12}{5} \cdot 32$ .

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Step 3.2.1.1

Multiply $- \frac{12}{5} \cdot 32$ .

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Step 3.2.1.1.1

Multiply $32$ by $- 1$ .

$4 x - 7 = - 32 (\frac{12}{5})$

Step 3.2.1.1.2

Combine $- 32$ and $\frac{12}{5}$ .

$4 x - 7 = \frac{- 32 \cdot 12}{5}$

Step 3.2.1.1.3

Multiply $- 32$ by $12$ .

$4 x - 7 = \frac{- 384}{5}$

Step 3.2.1.2

Move the negative in front of the fraction.

$4 x - 7 = - \frac{384}{5}$

Step 4

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

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Step 4.1

Add $7$ to both sides of the equation.

$4 x = - \frac{384}{5} + 7$

Step 4.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$4 x = - \frac{384}{5} + 7 \cdot \frac{5}{5}$

Step 4.3

Combine $7$ and $\frac{5}{5}$ .

$4 x = - \frac{384}{5} + \frac{7 \cdot 5}{5}$

Step 4.4

Combine the numerators over the common denominator.

$4 x = \frac{- 384 + 7 \cdot 5}{5}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $7$ by $5$ .

$4 x = \frac{- 384 + 35}{5}$

Step 4.5.2

Add $- 384$ and $35$ .

$4 x = \frac{- 349}{5}$

Step 4.6

Move the negative in front of the fraction.

$4 x = - \frac{349}{5}$

Step 5

Divide each term in $4 x = - \frac{349}{5}$ by $4$ and simplify.

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Step 5.1

Divide each term in $4 x = - \frac{349}{5}$ by $4$ .

$\frac{4 x}{4} = \frac{- \frac{349}{5}}{4}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $4$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- \frac{349}{5}}{4}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{349}{5}}{4}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{349}{5} \cdot \frac{1}{4}$

Step 5.3.2

Multiply $- \frac{349}{5} \cdot \frac{1}{4}$ .

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Step 5.3.2.1

Multiply $\frac{1}{4}$ by $\frac{349}{5}$ .

$x = - \frac{349}{4 \cdot 5}$

Step 5.3.2.2

Multiply $4$ by $5$ .

$x = - \frac{349}{20}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{349}{20}$

Decimal Form:

$x = - 17.45$

Mixed Number Form:

$x = - 17 \frac{9}{20}$