Algebra Examples

$\frac{x}{r - 8} + \frac{1}{8} = \frac{x + 1}{8 - r}$

Step 1

Multiply both sides by $8 - r$ .

$(\frac{x}{r - 8} + \frac{1}{8}) (8 - r) = \frac{x + 1}{8 - r} (8 - r)$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{x}{r - 8} + \frac{1}{8}) (8 - r)$ .

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Step 2.1.1.1

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

$(\frac{x}{r - 8} \cdot \frac{8}{8} + \frac{1}{8}) (8 - r) = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.2

To write $\frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{r - 8}{r - 8}$ .

$(\frac{x}{r - 8} \cdot \frac{8}{8} + \frac{1}{8} \cdot \frac{r - 8}{r - 8}) (8 - r) = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.3

Write each expression with a common denominator of $(r - 8) \cdot 8$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1.1.3.1

Multiply $\frac{x}{r - 8}$ by $\frac{8}{8}$ .

$(\frac{x \cdot 8}{(r - 8) \cdot 8} + \frac{1}{8} \cdot \frac{r - 8}{r - 8}) (8 - r) = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.3.2

Multiply $\frac{1}{8}$ by $\frac{r - 8}{r - 8}$ .

$(\frac{x \cdot 8}{(r - 8) \cdot 8} + \frac{r - 8}{8 (r - 8)}) (8 - r) = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.3.3

Reorder the factors of $(r - 8) \cdot 8$ .

$(\frac{x \cdot 8}{8 (r - 8)} + \frac{r - 8}{8 (r - 8)}) (8 - r) = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.4

Combine the numerators over the common denominator.

$\frac{x \cdot 8 + r - 8}{8 (r - 8)} (8 - r) = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.5

Move $8$ to the left of $x$ .

$\frac{8 x + r - 8}{8 (r - 8)} (8 - r) = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.6

Simplify terms.

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Step 2.1.1.6.1

Multiply $\frac{8 x + r - 8}{8 (r - 8)}$ by $8 - r$ .

$\frac{(8 x + r - 8) (8 - r)}{8 (r - 8)} = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.6.2

Cancel the common factor of $8 - r$ and $r - 8$ .

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Step 2.1.1.6.2.1

Rewrite $8$ as $- 1 (- 8)$ .

$\frac{(8 x + r - 8) (- 1 (- 8) - r)}{8 (r - 8)} = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.6.2.2

Factor $- 1$ out of $- r$ .

$\frac{(8 x + r - 8) (- 1 (- 8) - (r))}{8 (r - 8)} = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.6.2.3

Factor $- 1$ out of $- 1 (- 8) - (r)$ .

$\frac{(8 x + r - 8) (- 1 (- 8 + r))}{8 (r - 8)} = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.6.2.4

Reorder terms.

$\frac{(8 x + r - 8) (- 1 (r - 8))}{8 (r - 8)} = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.6.2.5

Cancel the common factor.

$\frac{(8 x + r - 8) (- 1 (r - 8))}{8 (r - 8)} = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.6.2.6

Rewrite the expression.

$\frac{(8 x + r - 8) \cdot (- 1)}{8} = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.6.3

Simplify the expression.

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Step 2.1.1.6.3.1

Move $- 1$ to the left of $8 x + r - 8$ .

$\frac{- 1 \cdot (8 x + r - 8)}{8} = \frac{x + 1}{8 - r} (8 - r)$

Step 2.1.1.6.3.2

Move the negative in front of the fraction.

$- \frac{8 x + r - 8}{8} = \frac{x + 1}{8 - r} (8 - r)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $8 - r$ .

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Step 2.2.1.1

Cancel the common factor.

$- \frac{8 x + r - 8}{8} = \frac{x + 1}{8 - r} (8 - r)$

Step 2.2.1.2

Rewrite the expression.

$- \frac{8 x + r - 8}{8} = x + 1$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $8$ .

$- \frac{8 x + r - 8}{8} \cdot 8 = (x + 1) \cdot 8$

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 $- \frac{8 x + r - 8}{8} \cdot 8$ .

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Step 3.2.1.1.1

Cancel the common factor of $8$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{8 x + r - 8}{8}$ into the numerator.

$\frac{- (8 x + r - 8)}{8} \cdot 8 = (x + 1) \cdot 8$

Step 3.2.1.1.1.2

Cancel the common factor.

$\frac{- (8 x + r - 8)}{8} \cdot 8 = (x + 1) \cdot 8$

Step 3.2.1.1.1.3

Rewrite the expression.

$- (8 x + r - 8) = (x + 1) \cdot 8$

Step 3.2.1.1.2

Apply the distributive property.

$- (8 x) - r - - 8 = (x + 1) \cdot 8$

Step 3.2.1.1.3

Simplify.

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Step 3.2.1.1.3.1

Multiply $8$ by $- 1$ .

$- 8 x - r - - 8 = (x + 1) \cdot 8$

Step 3.2.1.1.3.2

Multiply $- 1$ by $- 8$ .

$- 8 x - r + 8 = (x + 1) \cdot 8$

Step 3.2.1.1.4

Reorder $- 8 x$ and $- r$ .

$- r - 8 x + 8 = (x + 1) \cdot 8$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(x + 1) \cdot 8$ .

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Step 3.2.2.1.1

Apply the distributive property.

$- r - 8 x + 8 = x \cdot 8 + 1 \cdot 8$

Step 3.2.2.1.2

Simplify the expression.

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Step 3.2.2.1.2.1

Move $8$ to the left of $x$ .

$- r - 8 x + 8 = 8 \cdot x + 1 \cdot 8$

Step 3.2.2.1.2.2

Multiply $8$ by $1$ .

$- r - 8 x + 8 = 8 x + 8$

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 $8 x$ from both sides of the equation.

$- r - 8 x + 8 - 8 x = 8$

Step 3.3.1.2

Subtract $8 x$ from $- 8 x$ .

$- r - 16 x + 8 = 8$

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 $r$ to both sides of the equation.

$- 16 x + 8 = 8 + r$

Step 3.3.2.2

Subtract $8$ from both sides of the equation.

$- 16 x = 8 + r - 8$

Step 3.3.2.3

Combine the opposite terms in $8 + r - 8$ .

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Step 3.3.2.3.1

Subtract $8$ from $8$ .

$- 16 x = r + 0$

Step 3.3.2.3.2

Add $r$ and $0$ .

$- 16 x = r$

Step 3.3.3

Divide each term in $- 16 x = r$ by $- 16$ and simplify.

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Step 3.3.3.1

Divide each term in $- 16 x = r$ by $- 16$ .

$\frac{- 16 x}{- 16} = \frac{r}{- 16}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 16$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 16 x}{- 16} = \frac{r}{- 16}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{r}{- 16}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Move the negative in front of the fraction.

$x = - \frac{r}{16}$