Pre-Algebra Examples

$\frac{x - 7}{9} - \frac{x}{7} = \frac{7 x - 9}{6}$

Step 1

Multiply both sides by $6$ .

$(\frac{x - 7}{9} - \frac{x}{7}) \cdot 6 = \frac{7 x - 9}{6} \cdot 6$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{x - 7}{9} - \frac{x}{7}) \cdot 6$ .

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Step 2.1.1.1

To write $\frac{x - 7}{9}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$(\frac{x - 7}{9} \cdot \frac{7}{7} - \frac{x}{7}) \cdot 6 = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.2

To write $- \frac{x}{7}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$(\frac{x - 7}{9} \cdot \frac{7}{7} - \frac{x}{7} \cdot \frac{9}{9}) \cdot 6 = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.3

Write each expression with a common denominator of $63$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1.1.3.1

Multiply $\frac{x - 7}{9}$ by $\frac{7}{7}$ .

$(\frac{(x - 7) \cdot 7}{9 \cdot 7} - \frac{x}{7} \cdot \frac{9}{9}) \cdot 6 = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.3.2

Multiply $9$ by $7$ .

$(\frac{(x - 7) \cdot 7}{63} - \frac{x}{7} \cdot \frac{9}{9}) \cdot 6 = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.3.3

Multiply $\frac{x}{7}$ by $\frac{9}{9}$ .

$(\frac{(x - 7) \cdot 7}{63} - \frac{x \cdot 9}{7 \cdot 9}) \cdot 6 = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.3.4

Multiply $7$ by $9$ .

$(\frac{(x - 7) \cdot 7}{63} - \frac{x \cdot 9}{63}) \cdot 6 = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.4

Combine the numerators over the common denominator.

$\frac{(x - 7) \cdot 7 - x \cdot 9}{63} \cdot 6 = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.5

Simplify the numerator.

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Step 2.1.1.5.1

Apply the distributive property.

$\frac{x \cdot 7 - 7 \cdot 7 - x \cdot 9}{63} \cdot 6 = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.5.2

Move $7$ to the left of $x$ .

$\frac{7 \cdot x - 7 \cdot 7 - x \cdot 9}{63} \cdot 6 = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.5.3

Multiply $- 7$ by $7$ .

$\frac{7 \cdot x - 49 - x \cdot 9}{63} \cdot 6 = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.5.4

Multiply $9$ by $- 1$ .

$\frac{7 x - 49 - 9 x}{63} \cdot 6 = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.5.5

Subtract $9 x$ from $7 x$ .

$\frac{- 2 x - 49}{63} \cdot 6 = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.6

Simplify terms.

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Step 2.1.1.6.1

Cancel the common factor of $3$ .

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Step 2.1.1.6.1.1

Factor $3$ out of $63$ .

$\frac{- 2 x - 49}{3 (21)} \cdot 6 = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.6.1.2

Factor $3$ out of $6$ .

$\frac{- 2 x - 49}{3 \cdot 21} \cdot (3 \cdot 2) = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.6.1.3

Cancel the common factor.

$\frac{- 2 x - 49}{3 \cdot 21} \cdot (3 \cdot 2) = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.6.1.4

Rewrite the expression.

$\frac{- 2 x - 49}{21} \cdot 2 = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.6.2

Combine $\frac{- 2 x - 49}{21}$ and $2$ .

$\frac{(- 2 x - 49) \cdot 2}{21} = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.6.3

Move $2$ to the left of $- 2 x - 49$ .

$\frac{2 \cdot (- 2 x - 49)}{21} = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.6.4

Factor $- 1$ out of $- 2 x$ .

$\frac{2 (- (2 x) - 49)}{21} = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.6.5

Rewrite $- 49$ as $- 1 (49)$ .

$\frac{2 (- (2 x) - 1 (49))}{21} = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.6.6

Factor $- 1$ out of $- (2 x) - 1 (49)$ .

$\frac{2 (- (2 x + 49))}{21} = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.6.7

Simplify the expression.

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Step 2.1.1.6.7.1

Rewrite $- (2 x + 49)$ as $- 1 (2 x + 49)$ .

$\frac{2 (- 1 (2 x + 49))}{21} = \frac{7 x - 9}{6} \cdot 6$

Step 2.1.1.6.7.2

Move the negative in front of the fraction.

$- \frac{2 (2 x + 49)}{21} = \frac{7 x - 9}{6} \cdot 6$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $6$ .

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Step 2.2.1.1

Cancel the common factor.

$- \frac{2 (2 x + 49)}{21} = \frac{7 x - 9}{6} \cdot 6$

Step 2.2.1.2

Rewrite the expression.

$- \frac{2 (2 x + 49)}{21} = 7 x - 9$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $21$ .

$- \frac{2 (2 x + 49)}{21} \cdot 21 = (7 x - 9) \cdot 21$

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 $- \frac{2 (2 x + 49)}{21} \cdot 21$ .

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Step 3.2.1.1.1

Cancel the common factor of $21$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{2 (2 x + 49)}{21}$ into the numerator.

$\frac{- 2 (2 x + 49)}{21} \cdot 21 = (7 x - 9) \cdot 21$

Step 3.2.1.1.1.2

Cancel the common factor.

$\frac{- 2 (2 x + 49)}{21} \cdot 21 = (7 x - 9) \cdot 21$

Step 3.2.1.1.1.3

Rewrite the expression.

$- 2 (2 x + 49) = (7 x - 9) \cdot 21$

Step 3.2.1.1.2

Apply the distributive property.

$- 2 (2 x) - 2 \cdot 49 = (7 x - 9) \cdot 21$

Step 3.2.1.1.3

Multiply.

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Step 3.2.1.1.3.1

Multiply $2$ by $- 2$ .

$- 4 x - 2 \cdot 49 = (7 x - 9) \cdot 21$

Step 3.2.1.1.3.2

Multiply $- 2$ by $49$ .

$- 4 x - 98 = (7 x - 9) \cdot 21$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(7 x - 9) \cdot 21$ .

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Step 3.2.2.1.1

Apply the distributive property.

$- 4 x - 98 = 7 x \cdot 21 - 9 \cdot 21$

Step 3.2.2.1.2

Multiply.

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Step 3.2.2.1.2.1

Multiply $21$ by $7$ .

$- 4 x - 98 = 147 x - 9 \cdot 21$

Step 3.2.2.1.2.2

Multiply $- 9$ by $21$ .

$- 4 x - 98 = 147 x - 189$

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

Subtract $147 x$ from both sides of the equation.

$- 4 x - 98 - 147 x = - 189$

Step 3.3.1.2

Subtract $147 x$ from $- 4 x$ .

$- 151 x - 98 = - 189$

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

Add $98$ to both sides of the equation.

$- 151 x = - 189 + 98$

Step 3.3.2.2

Add $- 189$ and $98$ .

$- 151 x = - 91$

Step 3.3.3

Divide each term in $- 151 x = - 91$ by $- 151$ and simplify.

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Step 3.3.3.1

Divide each term in $- 151 x = - 91$ by $- 151$ .

$\frac{- 151 x}{- 151} = \frac{- 91}{- 151}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 151$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 151 x}{- 151} = \frac{- 91}{- 151}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 91}{- 151}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{91}{151}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{91}{151}$

Decimal Form:

$x = 0.60264900 \dots$