Find the Equation Using Point-Slope Formula (45,169) , (65,173)

$(45, 169)$ , $(65, 173)$

Step 1

Find the slope of the line between $(45, 169)$ and $(65, 173)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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

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

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).

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

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

$m = \frac{173 - (169)}{65 - (45)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $169$ .

$m = \frac{173 - 169}{65 - (45)}$

Subtract $169$ from $173$ .

$m = \frac{4}{65 - (45)}$

Simplify the denominator.

Multiply $- 1$ by $45$ .

$m = \frac{4}{65 - 45}$

Subtract $45$ from $65$ .

$m = \frac{4}{20}$

Cancel the common factor of $4$ and $20$ .

Factor $4$ out of $4$ .

$m = \frac{4 (1)}{20}$

Cancel the common factors.

Factor $4$ out of $20$ .

$m = \frac{4 \cdot 1}{4 \cdot 5}$

Cancel the common factor.

$m = \frac{4 \cdot 1}{4 \cdot 5}$

Rewrite the expression.

$m = \frac{1}{5}$

Step 2

Use the slope $\frac{1}{5}$ and a given point $(45, 169)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (169) = \frac{1}{5} \cdot (x - (45))$

Step 3

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

$y - 169 = \frac{1}{5} \cdot (x - 45)$

Step 4

Solve for $y$ .

Simplify $\frac{1}{5} \cdot (x - 45)$ .

Rewrite.

$y - 169 = 0 + 0 + \frac{1}{5} \cdot (x - 45)$

Simplify by adding zeros.

$y - 169 = \frac{1}{5} \cdot (x - 45)$

Apply the distributive property.

$y - 169 = \frac{1}{5} x + \frac{1}{5} \cdot - 45$

Combine $\frac{1}{5}$ and $x$ .

$y - 169 = \frac{x}{5} + \frac{1}{5} \cdot - 45$

Cancel the common factor of $5$ .

Factor $5$ out of $- 45$ .

$y - 169 = \frac{x}{5} + \frac{1}{5} \cdot (5 (- 9))$

Cancel the common factor.

$y - 169 = \frac{x}{5} + \frac{1}{5} \cdot (5 \cdot - 9)$

Rewrite the expression.

$y - 169 = \frac{x}{5} - 9$

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

Add $169$ to both sides of the equation.

$y = \frac{x}{5} - 9 + 169$

Add $- 9$ and $169$ .

$y = \frac{x}{5} + 160$

Step 5

Reorder terms.

$y = \frac{1}{5} x + 160$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{1}{5} x + 160$

Point-slope form:

$y - 169 = \frac{1}{5} \cdot (x - 45)$

Step 7

Find the Equation Using Point-Slope Formula (45,169) , (65,173)

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