Find the Equation Using Point-Slope Formula (10,3) , (6,31)

$(10, 3)$ , $(6, 31)$

Step 1

Find the slope of the line between $(10, 3)$ and $(6, 31)$ 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.

$m = \frac{31 - (3)}{6 - (10)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $3$ .

$m = \frac{31 - 3}{6 - (10)}$

Subtract $3$ from $31$ .

$m = \frac{28}{6 - (10)}$

Simplify the denominator.

Multiply $- 1$ by $10$ .

$m = \frac{28}{6 - 10}$

Subtract $10$ from $6$ .

$m = \frac{28}{- 4}$

Divide $28$ by $- 4$ .

$m = - 7$

Step 2

Use the slope $- 7$ and a given point $(10, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = - 7 \cdot (x - (10))$

Step 3

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

$y - 3 = - 7 \cdot (x - 10)$

Step 4

Solve for $y$ .

Simplify $- 7 \cdot (x - 10)$ .

Rewrite.

$y - 3 = 0 + 0 - 7 \cdot (x - 10)$

Simplify by adding zeros.

$y - 3 = - 7 \cdot (x - 10)$

Apply the distributive property.

$y - 3 = - 7 x - 7 \cdot - 10$

Multiply $- 7$ by $- 10$ .

$y - 3 = - 7 x + 70$

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

Add $3$ to both sides of the equation.

$y = - 7 x + 70 + 3$

Add $70$ and $3$ .

$y = - 7 x + 73$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = - 7 x + 73$

Point-slope form:

$y - 3 = - 7 \cdot (x - 10)$

Step 6

Find the Equation Using Point-Slope Formula (10,3) , (6,31)

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