Finite Math Examples

$(8, - 3)$ , $(14, 12)$

Step 1

Find the slope of the line between $(8, - 3)$ and $(14, 12)$ 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.

$m = \frac{12 - (- 3)}{14 - (8)}$

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 $- 3$ .

$m = \frac{12 + 3}{14 - (8)}$

Step 1.4.1.2

Add $12$ and $3$ .

$m = \frac{15}{14 - (8)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $8$ .

$m = \frac{15}{14 - 8}$

Step 1.4.2.2

Subtract $8$ from $14$ .

$m = \frac{15}{6}$

Step 1.4.3

Cancel the common factor of $15$ and $6$ .

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Step 1.4.3.1

Factor $3$ out of $15$ .

$m = \frac{3 (5)}{6}$

Step 1.4.3.2

Cancel the common factors.

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Step 1.4.3.2.1

Factor $3$ out of $6$ .

$m = \frac{3 \cdot 5}{3 \cdot 2}$

Step 1.4.3.2.2

Cancel the common factor.

$m = \frac{3 \cdot 5}{3 \cdot 2}$

Step 1.4.3.2.3

Rewrite the expression.

$m = \frac{5}{2}$

Step 2

Use the slope $\frac{5}{2}$ and a given point $(8, - 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 3) = \frac{5}{2} \cdot (x - (8))$

Step 3

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

$y + 3 = \frac{5}{2} \cdot (x - 8)$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $\frac{5}{2} \cdot (x - 8)$ .

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Step 4.1.1

Rewrite.

$y + 3 = 0 + 0 + \frac{5}{2} \cdot (x - 8)$

Step 4.1.2

Simplify by adding zeros.

$y + 3 = \frac{5}{2} \cdot (x - 8)$

Step 4.1.3

Apply the distributive property.

$y + 3 = \frac{5}{2} x + \frac{5}{2} \cdot - 8$

Step 4.1.4

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

$y + 3 = \frac{5 x}{2} + \frac{5}{2} \cdot - 8$

Step 4.1.5

Cancel the common factor of $2$ .

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Step 4.1.5.1

Factor $2$ out of $- 8$ .

$y + 3 = \frac{5 x}{2} + \frac{5}{2} \cdot (2 (- 4))$

Step 4.1.5.2

Cancel the common factor.

$y + 3 = \frac{5 x}{2} + \frac{5}{2} \cdot (2 \cdot - 4)$

Step 4.1.5.3

Rewrite the expression.

$y + 3 = \frac{5 x}{2} + 5 \cdot - 4$

Step 4.1.6

Multiply $5$ by $- 4$ .

$y + 3 = \frac{5 x}{2} - 20$

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.

$y = \frac{5 x}{2} - 20 - 3$

Step 4.2.2

Subtract $3$ from $- 20$ .

$y = \frac{5 x}{2} - 23$

Step 5

Reorder terms.

$y = \frac{5}{2} x - 23$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{5}{2} x - 23$

Point-slope form:

$y + 3 = \frac{5}{2} \cdot (x - 8)$

Step 7