Algebra Examples

$(3, 5)$ , $(- 3, r)$ , $m = \frac{3}{4}$

Step 1

Slope $m$ is equal to the change in $y$ over the change in $x$ .

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 2

Substitute in the values of the known parameters.

$\frac{3}{4} = \frac{r - (5)}{- 3 - (3)}$

Step 3

Solve for $r$ .

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Step 3.1

Rewrite the equation as $\frac{r - (5)}{- 3 - (3)} = \frac{3}{4}$ .

$\frac{r - (5)}{- 3 - (3)} = \frac{3}{4}$

Step 3.2

Multiply both sides of the equation by $- 3 - (3)$ .

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

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $(- 3 - 1 \cdot 3) \frac{r - (5)}{- 3 - 1 \cdot 3}$ .

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Step 3.3.1.1.1

Multiply $- 1$ by $5$ .

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

Step 3.3.1.1.2

Simplify the denominator.

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Step 3.3.1.1.2.1

Multiply $- 1$ by $3$ .

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

Step 3.3.1.1.2.2

Subtract $3$ from $- 3$ .

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

Step 3.3.1.1.3

Simplify terms.

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Step 3.3.1.1.3.1

Multiply $- 1$ by $3$ .

$(- 3 - 3) \frac{r - 5}{- 6} = (- 3 - (3)) \frac{3}{4}$

Step 3.3.1.1.3.2

Subtract $3$ from $- 3$ .

$- 6 \frac{r - 5}{- 6} = (- 3 - (3)) \frac{3}{4}$

Step 3.3.1.1.3.3

Cancel the common factor of $6$ .

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Step 3.3.1.1.3.3.1

Factor $6$ out of $- 6$ .

$6 (- 1) \frac{r - 5}{- 6} = (- 3 - (3)) \frac{3}{4}$

Step 3.3.1.1.3.3.2

Factor $6$ out of $- 6$ .

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

Step 3.3.1.1.3.3.3

Cancel the common factor.

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

Step 3.3.1.1.3.3.4

Rewrite the expression.

$- 1 \frac{r - 5}{- 1} = (- 3 - (3)) \frac{3}{4}$

Step 3.3.1.1.3.4

Simplify the expression.

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Step 3.3.1.1.3.4.1

Move the negative one from the denominator of $\frac{r - 5}{- 1}$ .

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

Step 3.3.1.1.3.4.2

Rewrite $- 1 \cdot (r - 5)$ as $- (r - 5)$ .

$- 1 (- (r - 5)) = (- 3 - (3)) \frac{3}{4}$

Step 3.3.1.1.3.5

Apply the distributive property.

$- 1 (- r - - 5) = (- 3 - (3)) \frac{3}{4}$

Step 3.3.1.1.3.6

Multiply $- 1$ by $- 5$ .

$- 1 (- r + 5) = (- 3 - (3)) \frac{3}{4}$

Step 3.3.1.1.3.7

Apply the distributive property.

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

Step 3.3.1.1.4

Multiply $- 1 (- r)$ .

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Step 3.3.1.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.1.4.2

Multiply $r$ by $1$ .

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

Step 3.3.1.1.5

Multiply $- 1$ by $5$ .

$r - 5 = (- 3 - (3)) \frac{3}{4}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $(- 3 - (3)) \frac{3}{4}$ .

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Step 3.3.2.1.1

Simplify the expression.

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Step 3.3.2.1.1.1

Multiply $- 1$ by $3$ .

$r - 5 = (- 3 - 3) \frac{3}{4}$

Step 3.3.2.1.1.2

Subtract $3$ from $- 3$ .

$r - 5 = - 6 (\frac{3}{4})$

Step 3.3.2.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.2.1

Factor $2$ out of $- 6$ .

$r - 5 = 2 (- 3) \frac{3}{4}$

Step 3.3.2.1.2.2

Factor $2$ out of $4$ .

$r - 5 = 2 \cdot - 3 \frac{3}{2 \cdot 2}$

Step 3.3.2.1.2.3

Cancel the common factor.

$r - 5 = 2 \cdot - 3 \frac{3}{2 \cdot 2}$

Step 3.3.2.1.2.4

Rewrite the expression.

$r - 5 = - 3 (\frac{3}{2})$

Step 3.3.2.1.3

Combine $- 3$ and $\frac{3}{2}$ .

$r - 5 = \frac{- 3 \cdot 3}{2}$

Step 3.3.2.1.4

Simplify the expression.

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Step 3.3.2.1.4.1

Multiply $- 3$ by $3$ .

$r - 5 = \frac{- 9}{2}$

Step 3.3.2.1.4.2

Move the negative in front of the fraction.

$r - 5 = - \frac{9}{2}$

Step 3.4

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

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Step 3.4.1

Add $5$ to both sides of the equation.

$r = - \frac{9}{2} + 5$

Step 3.4.2

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

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

Step 3.4.3

Combine $5$ and $\frac{2}{2}$ .

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

Step 3.4.4

Combine the numerators over the common denominator.

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

Step 3.4.5

Simplify the numerator.

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Step 3.4.5.1

Multiply $5$ by $2$ .

$r = \frac{- 9 + 10}{2}$

Step 3.4.5.2

Add $- 9$ and $10$ .

$r = \frac{1}{2}$