Pre-Algebra Examples

$(13, - 3)$ , $(6, - 11)$

Step 1

Find the slope of the line between $(13, - 3)$ and $(6, - 11)$ 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.

$m = \frac{- 11 - (- 3)}{6 - (13)}$

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 $- 3$ .

$m = \frac{- 11 + 3}{6 - (13)}$

Step 1.4.1.2

Add $- 11$ and $3$ .

$m = \frac{- 8}{6 - (13)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $13$ .

$m = \frac{- 8}{6 - 13}$

Step 1.4.2.2

Subtract $13$ from $6$ .

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

Step 1.4.3

Dividing two negative values results in a positive value.

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

Step 2

Use the slope $\frac{8}{7}$ and a given point $(13, - 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 3) = \frac{8}{7} \cdot (x - (13))$

Step 3

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

$y + 3 = \frac{8}{7} \cdot (x - 13)$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $\frac{8}{7} \cdot (x - 13)$ .

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Step 4.1.1

Rewrite.

$y + 3 = 0 + 0 + \frac{8}{7} \cdot (x - 13)$

Step 4.1.2

Simplify by adding zeros.

$y + 3 = \frac{8}{7} \cdot (x - 13)$

Step 4.1.3

Apply the distributive property.

$y + 3 = \frac{8}{7} x + \frac{8}{7} \cdot - 13$

Step 4.1.4

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

$y + 3 = \frac{8 x}{7} + \frac{8}{7} \cdot - 13$

Step 4.1.5

Multiply $\frac{8}{7} \cdot - 13$ .

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Step 4.1.5.1

Combine $\frac{8}{7}$ and $- 13$ .

$y + 3 = \frac{8 x}{7} + \frac{8 \cdot - 13}{7}$

Step 4.1.5.2

Multiply $8$ by $- 13$ .

$y + 3 = \frac{8 x}{7} + \frac{- 104}{7}$

Step 4.1.6

Move the negative in front of the fraction.

$y + 3 = \frac{8 x}{7} - \frac{104}{7}$

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 $3$ from both sides of the equation.

$y = \frac{8 x}{7} - \frac{104}{7} - 3$

Step 4.2.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = \frac{8 x}{7} - \frac{104}{7} - 3 \cdot \frac{7}{7}$

Step 4.2.3

Combine $- 3$ and $\frac{7}{7}$ .

$y = \frac{8 x}{7} - \frac{104}{7} + \frac{- 3 \cdot 7}{7}$

Step 4.2.4

Combine the numerators over the common denominator.

$y = \frac{8 x}{7} + \frac{- 104 - 3 \cdot 7}{7}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 3$ by $7$ .

$y = \frac{8 x}{7} + \frac{- 104 - 21}{7}$

Step 4.2.5.2

Subtract $21$ from $- 104$ .

$y = \frac{8 x}{7} + \frac{- 125}{7}$

Step 4.2.6

Move the negative in front of the fraction.

$y = \frac{8 x}{7} - \frac{125}{7}$

Step 5

Reorder terms.

$y = \frac{8}{7} x - \frac{125}{7}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{8}{7} x - \frac{125}{7}$

Point-slope form:

$y + 3 = \frac{8}{7} \cdot (x - 13)$

Step 7