Finite Math Examples

$(- 2, 5)$ , $(- 5, - 9)$

Step 1

Find the slope of the line between $(- 2, 5)$ and $(- 5, - 9)$ 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.

$m = \frac{- 9 - (5)}{- 5 - (- 2)}$

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 $5$ .

$m = \frac{- 9 - 5}{- 5 - (- 2)}$

Step 1.4.1.2

Subtract $5$ from $- 9$ .

$m = \frac{- 14}{- 5 - (- 2)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $- 2$ .

$m = \frac{- 14}{- 5 + 2}$

Step 1.4.2.2

Add $- 5$ and $2$ .

$m = \frac{- 14}{- 3}$

Step 1.4.3

Dividing two negative values results in a positive value.

$m = \frac{14}{3}$

Step 2

Use the slope $\frac{14}{3}$ and a given point $(- 2, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = \frac{14}{3} \cdot (x - (- 2))$

Step 3

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

$y - 5 = \frac{14}{3} \cdot (x + 2)$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $\frac{14}{3} \cdot (x + 2)$ .

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Step 4.1.1

Rewrite.

$y - 5 = 0 + 0 + \frac{14}{3} \cdot (x + 2)$

Step 4.1.2

Simplify by adding zeros.

$y - 5 = \frac{14}{3} \cdot (x + 2)$

Step 4.1.3

Apply the distributive property.

$y - 5 = \frac{14}{3} x + \frac{14}{3} \cdot 2$

Step 4.1.4

Combine $\frac{14}{3}$ and $x$ .

$y - 5 = \frac{14 x}{3} + \frac{14}{3} \cdot 2$

Step 4.1.5

Multiply $\frac{14}{3} \cdot 2$ .

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Step 4.1.5.1

Combine $\frac{14}{3}$ and $2$ .

$y - 5 = \frac{14 x}{3} + \frac{14 \cdot 2}{3}$

Step 4.1.5.2

Multiply $14$ by $2$ .

$y - 5 = \frac{14 x}{3} + \frac{28}{3}$

Step 4.2

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

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Step 4.2.1

Add $5$ to both sides of the equation.

$y = \frac{14 x}{3} + \frac{28}{3} + 5$

Step 4.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{14 x}{3} + \frac{28}{3} + 5 \cdot \frac{3}{3}$

Step 4.2.3

Combine $5$ and $\frac{3}{3}$ .

$y = \frac{14 x}{3} + \frac{28}{3} + \frac{5 \cdot 3}{3}$

Step 4.2.4

Combine the numerators over the common denominator.

$y = \frac{14 x}{3} + \frac{28 + 5 \cdot 3}{3}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $5$ by $3$ .

$y = \frac{14 x}{3} + \frac{28 + 15}{3}$

Step 4.2.5.2

Add $28$ and $15$ .

$y = \frac{14 x}{3} + \frac{43}{3}$

Step 5

Reorder terms.

$y = \frac{14}{3} x + \frac{43}{3}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{14}{3} x + \frac{43}{3}$

Point-slope form:

$y - 5 = \frac{14}{3} \cdot (x + 2)$

Step 7