Algebra Examples

$(- 5, - 4)$ , $(- 13, - 11)$

Step 1

Find the slope of the line between $(- 5, - 4)$ and $(- 13, - 11)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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Step 1.1

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 1.2

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

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

Step 1.3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{- 11 - (- 4)}{- 13 - (- 5)}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Multiply $- 1$ by $- 4$ .

$m = \frac{- 11 + 4}{- 13 - (- 5)}$

Step 1.4.1.2

Add $- 11$ and $4$ .

$m = \frac{- 7}{- 13 - (- 5)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $- 5$ .

$m = \frac{- 7}{- 13 + 5}$

Step 1.4.2.2

Add $- 13$ and $5$ .

$m = \frac{- 7}{- 8}$

Step 1.4.3

Dividing two negative values results in a positive value.

$m = \frac{7}{8}$

Step 2

Use the slope $\frac{7}{8}$ and a given point $(- 5, - 4)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 4) = \frac{7}{8} \cdot (x - (- 5))$

Step 3

Simplify the equation and keep it in point-slope form.

$y + 4 = \frac{7}{8} \cdot (x + 5)$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $\frac{7}{8} \cdot (x + 5)$ .

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Step 4.1.1

Rewrite.

$y + 4 = 0 + 0 + \frac{7}{8} \cdot (x + 5)$

Step 4.1.2

Simplify by adding zeros.

$y + 4 = \frac{7}{8} \cdot (x + 5)$

Step 4.1.3

Apply the distributive property.

$y + 4 = \frac{7}{8} x + \frac{7}{8} \cdot 5$

Step 4.1.4

Combine $\frac{7}{8}$ and $x$ .

$y + 4 = \frac{7 x}{8} + \frac{7}{8} \cdot 5$

Step 4.1.5

Multiply $\frac{7}{8} \cdot 5$ .

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Step 4.1.5.1

Combine $\frac{7}{8}$ and $5$ .

$y + 4 = \frac{7 x}{8} + \frac{7 \cdot 5}{8}$

Step 4.1.5.2

Multiply $7$ by $5$ .

$y + 4 = \frac{7 x}{8} + \frac{35}{8}$

Step 4.2

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

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Step 4.2.1

Subtract $4$ from both sides of the equation.

$y = \frac{7 x}{8} + \frac{35}{8} - 4$

Step 4.2.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = \frac{7 x}{8} + \frac{35}{8} - 4 \cdot \frac{8}{8}$

Step 4.2.3

Combine $- 4$ and $\frac{8}{8}$ .

$y = \frac{7 x}{8} + \frac{35}{8} + \frac{- 4 \cdot 8}{8}$

Step 4.2.4

Combine the numerators over the common denominator.

$y = \frac{7 x}{8} + \frac{35 - 4 \cdot 8}{8}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 4$ by $8$ .

$y = \frac{7 x}{8} + \frac{35 - 32}{8}$

Step 4.2.5.2

Subtract $32$ from $35$ .

$y = \frac{7 x}{8} + \frac{3}{8}$

Step 5

Reorder terms.

$y = \frac{7}{8} x + \frac{3}{8}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{7}{8} x + \frac{3}{8}$

Point-slope form:

$y + 4 = \frac{7}{8} \cdot (x + 5)$

Step 7