Finite Math Examples

$(5, 8)$ , $(7, 5)$

Step 1

Find the slope of the line between $(5, 8)$ and $(7, 5)$ 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.

$m = \frac{5 - (8)}{7 - (5)}$

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 $8$ .

$m = \frac{5 - 8}{7 - (5)}$

Step 1.4.1.2

Subtract $8$ from $5$ .

$m = \frac{- 3}{7 - (5)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $5$ .

$m = \frac{- 3}{7 - 5}$

Step 1.4.2.2

Subtract $5$ from $7$ .

$m = \frac{- 3}{2}$

Step 1.4.3

Move the negative in front of the fraction.

$m = - \frac{3}{2}$

Step 2

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

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

Step 3

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

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

Step 4

Solve for $y$ .

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Step 4.1

Simplify $- \frac{3}{2} \cdot (x - 5)$ .

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Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by adding zeros.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

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

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

Step 4.1.5

Multiply $- \frac{3}{2} \cdot - 5$ .

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Step 4.1.5.1

Multiply $- 5$ by $- 1$ .

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

Step 4.1.5.2

Combine $5$ and $\frac{3}{2}$ .

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

Step 4.1.5.3

Multiply $5$ by $3$ .

$y - 8 = - \frac{x \cdot 3}{2} + \frac{15}{2}$

Step 4.1.6

Move $3$ to the left of $x$ .

$y - 8 = - \frac{3 x}{2} + \frac{15}{2}$

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 $8$ to both sides of the equation.

$y = - \frac{3 x}{2} + \frac{15}{2} + 8$

Step 4.2.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{3 x}{2} + \frac{15}{2} + 8 \cdot \frac{2}{2}$

Step 4.2.3

Combine $8$ and $\frac{2}{2}$ .

$y = - \frac{3 x}{2} + \frac{15}{2} + \frac{8 \cdot 2}{2}$

Step 4.2.4

Combine the numerators over the common denominator.

$y = - \frac{3 x}{2} + \frac{15 + 8 \cdot 2}{2}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $8$ by $2$ .

$y = - \frac{3 x}{2} + \frac{15 + 16}{2}$

Step 4.2.5.2

Add $15$ and $16$ .

$y = - \frac{3 x}{2} + \frac{31}{2}$

Step 5

Reorder terms.

$y = - (\frac{3}{2} x) + \frac{31}{2}$

Step 6

Remove parentheses.

$y = - \frac{3}{2} x + \frac{31}{2}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{3}{2} x + \frac{31}{2}$

Point-slope form:

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

Step 8