Algebra Examples

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

Step 1

Multiply both sides by $5$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{x - 1}{2} - \frac{x - 2}{3} - \frac{x - 3}{4}) \cdot 5$ .

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Step 2.1.1.1

Find the common denominator.

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Step 2.1.1.1.1

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

$(\frac{x - 1}{2} \cdot \frac{6}{6} - \frac{x - 2}{3} - \frac{x - 3}{4}) \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.1.2

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

$(\frac{(x - 1) \cdot 6}{2 \cdot 6} - \frac{x - 2}{3} - \frac{x - 3}{4}) \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.1.3

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

$(\frac{(x - 1) \cdot 6}{2 \cdot 6} - (\frac{x - 2}{3} \cdot \frac{4}{4}) - \frac{x - 3}{4}) \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.1.4

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

$(\frac{(x - 1) \cdot 6}{2 \cdot 6} - \frac{(x - 2) \cdot 4}{3 \cdot 4} - \frac{x - 3}{4}) \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.1.5

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

$(\frac{(x - 1) \cdot 6}{2 \cdot 6} - \frac{(x - 2) \cdot 4}{3 \cdot 4} - (\frac{x - 3}{4} \cdot \frac{3}{3})) \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.1.6

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

$(\frac{(x - 1) \cdot 6}{2 \cdot 6} - \frac{(x - 2) \cdot 4}{3 \cdot 4} - \frac{(x - 3) \cdot 3}{4 \cdot 3}) \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.1.7

Multiply $2$ by $6$ .

$(\frac{(x - 1) \cdot 6}{12} - \frac{(x - 2) \cdot 4}{3 \cdot 4} - \frac{(x - 3) \cdot 3}{4 \cdot 3}) \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.1.8

Multiply $3$ by $4$ .

$(\frac{(x - 1) \cdot 6}{12} - \frac{(x - 2) \cdot 4}{12} - \frac{(x - 3) \cdot 3}{4 \cdot 3}) \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.1.9

Reorder the factors of $4 \cdot 3$ .

$(\frac{(x - 1) \cdot 6}{12} - \frac{(x - 2) \cdot 4}{12} - \frac{(x - 3) \cdot 3}{3 \cdot 4}) \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.1.10

Multiply $3$ by $4$ .

$(\frac{(x - 1) \cdot 6}{12} - \frac{(x - 2) \cdot 4}{12} - \frac{(x - 3) \cdot 3}{12}) \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.2

Combine the numerators over the common denominator.

$\frac{(x - 1) \cdot 6 - (x - 2) \cdot 4 - (x - 3) \cdot 3}{12} \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.3

Simplify each term.

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Step 2.1.1.3.1

Apply the distributive property.

$\frac{x \cdot 6 - 1 \cdot 6 - (x - 2) \cdot 4 - (x - 3) \cdot 3}{12} \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.3.2

Move $6$ to the left of $x$ .

$\frac{6 \cdot x - 1 \cdot 6 - (x - 2) \cdot 4 - (x - 3) \cdot 3}{12} \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.3.3

Multiply $- 1$ by $6$ .

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

Step 2.1.1.3.4

Apply the distributive property.

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

Step 2.1.1.3.5

Multiply $- 1$ by $- 2$ .

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

Step 2.1.1.3.6

Apply the distributive property.

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

Step 2.1.1.3.7

Multiply $4$ by $- 1$ .

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

Step 2.1.1.3.8

Multiply $2$ by $4$ .

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

Step 2.1.1.3.9

Apply the distributive property.

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

Step 2.1.1.3.10

Multiply $- 1$ by $- 3$ .

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

Step 2.1.1.3.11

Apply the distributive property.

$\frac{6 x - 6 - 4 x + 8 - x \cdot 3 + 3 \cdot 3}{12} \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.3.12

Multiply $3$ by $- 1$ .

$\frac{6 x - 6 - 4 x + 8 - 3 x + 3 \cdot 3}{12} \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.3.13

Multiply $3$ by $3$ .

$\frac{6 x - 6 - 4 x + 8 - 3 x + 9}{12} \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.4

Simplify terms.

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Step 2.1.1.4.1

Subtract $4 x$ from $6 x$ .

$\frac{2 x - 6 + 8 - 3 x + 9}{12} \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.4.2

Subtract $3 x$ from $2 x$ .

$\frac{- x - 6 + 8 + 9}{12} \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.4.3

Simplify by adding numbers.

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Step 2.1.1.4.3.1

Add $- 6$ and $8$ .

$\frac{- x + 2 + 9}{12} \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.4.3.2

Add $2$ and $9$ .

$\frac{- x + 11}{12} \cdot 5 = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.4.4

Combine $\frac{- x + 11}{12}$ and $5$ .

$\frac{(- x + 11) \cdot 5}{12} = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.4.5

Move $5$ to the left of $- x + 11$ .

$\frac{5 \cdot (- x + 11)}{12} = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.4.6

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

$\frac{5 (- (x) + 11)}{12} = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.4.7

Rewrite $11$ as $- 1 (- 11)$ .

$\frac{5 (- (x) - 1 (- 11))}{12} = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.4.8

Factor $- 1$ out of $- (x) - 1 (- 11)$ .

$\frac{5 (- (x - 11))}{12} = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.4.9

Simplify the expression.

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Step 2.1.1.4.9.1

Rewrite $- (x - 11)$ as $- 1 (x - 11)$ .

$\frac{5 (- 1 (x - 11))}{12} = - \frac{x - 5}{5} \cdot 5$

Step 2.1.1.4.9.2

Move the negative in front of the fraction.

$- \frac{5 (x - 11)}{12} = - \frac{x - 5}{5} \cdot 5$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $- \frac{x - 5}{5} \cdot 5$ .

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Step 2.2.1.1

Cancel the common factor of $5$ .

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Step 2.2.1.1.1

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

$- \frac{5 (x - 11)}{12} = \frac{- (x - 5)}{5} \cdot 5$

Step 2.2.1.1.2

Cancel the common factor.

$- \frac{5 (x - 11)}{12} = \frac{- (x - 5)}{5} \cdot 5$

Step 2.2.1.1.3

Rewrite the expression.

$- \frac{5 (x - 11)}{12} = - (x - 5)$

Step 2.2.1.2

Apply the distributive property.

$- \frac{5 (x - 11)}{12} = - x - - 5$

Step 2.2.1.3

Multiply $- 1$ by $- 5$ .

$- \frac{5 (x - 11)}{12} = - x + 5$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $12$ .

$- \frac{5 (x - 11)}{12} \cdot 12 = (- x + 5) \cdot 12$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $- \frac{5 (x - 11)}{12} \cdot 12$ .

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Step 3.2.1.1.1

Cancel the common factor of $12$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{5 (x - 11)}{12}$ into the numerator.

$\frac{- 5 (x - 11)}{12} \cdot 12 = (- x + 5) \cdot 12$

Step 3.2.1.1.1.2

Cancel the common factor.

$\frac{- 5 (x - 11)}{12} \cdot 12 = (- x + 5) \cdot 12$

Step 3.2.1.1.1.3

Rewrite the expression.

$- 5 (x - 11) = (- x + 5) \cdot 12$

Step 3.2.1.1.2

Apply the distributive property.

$- 5 x - 5 \cdot - 11 = (- x + 5) \cdot 12$

Step 3.2.1.1.3

Multiply $- 5$ by $- 11$ .

$- 5 x + 55 = (- x + 5) \cdot 12$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(- x + 5) \cdot 12$ .

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Step 3.2.2.1.1

Apply the distributive property.

$- 5 x + 55 = - x \cdot 12 + 5 \cdot 12$

Step 3.2.2.1.2

Multiply.

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Step 3.2.2.1.2.1

Multiply $12$ by $- 1$ .

$- 5 x + 55 = - 12 x + 5 \cdot 12$

Step 3.2.2.1.2.2

Multiply $5$ by $12$ .

$- 5 x + 55 = - 12 x + 60$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

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Step 3.3.1.1

Add $12 x$ to both sides of the equation.

$- 5 x + 55 + 12 x = 60$

Step 3.3.1.2

Add $- 5 x$ and $12 x$ .

$7 x + 55 = 60$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $55$ from both sides of the equation.

$7 x = 60 - 55$

Step 3.3.2.2

Subtract $55$ from $60$ .

$7 x = 5$

Step 3.3.3

Divide each term in $7 x = 5$ by $7$ and simplify.

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Step 3.3.3.1

Divide each term in $7 x = 5$ by $7$ .

$\frac{7 x}{7} = \frac{5}{7}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $7$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{5}{7}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{7}$

Decimal Form:

$x = 0.\overline{714285}$