Find the Equation Using Point-Slope Formula (4,-2) , (6,5)

$(4, - 2)$ , $(6, 5)$

Step 1

Find the slope of the line between $(4, - 2)$ and $(6, 5)$ 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.

$m = \frac{5 - (- 2)}{6 - (4)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 2$ .

$m = \frac{5 + 2}{6 - (4)}$

Add $5$ and $2$ .

$m = \frac{7}{6 - (4)}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$m = \frac{7}{6 - 4}$

Subtract $4$ from $6$ .

$m = \frac{7}{2}$

Step 2

Use the slope $\frac{7}{2}$ and a given point $(4, - 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2) = \frac{7}{2} \cdot (x - (4))$

Step 3

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

$y + 2 = \frac{7}{2} \cdot (x - 4)$

Step 4

Solve for $y$ .

Simplify $\frac{7}{2} \cdot (x - 4)$ .

Rewrite.

$y + 2 = 0 + 0 + \frac{7}{2} \cdot (x - 4)$

Simplify by adding zeros.

$y + 2 = \frac{7}{2} \cdot (x - 4)$

Apply the distributive property.

$y + 2 = \frac{7}{2} x + \frac{7}{2} \cdot - 4$

Combine $\frac{7}{2}$ and $x$ .

$y + 2 = \frac{7 x}{2} + \frac{7}{2} \cdot - 4$

Cancel the common factor of $2$ .

Factor $2$ out of $- 4$ .

$y + 2 = \frac{7 x}{2} + \frac{7}{2} \cdot (2 (- 2))$

Cancel the common factor.

$y + 2 = \frac{7 x}{2} + \frac{7}{2} \cdot (2 \cdot - 2)$

Rewrite the expression.

$y + 2 = \frac{7 x}{2} + 7 \cdot - 2$

Multiply $7$ by $- 2$ .

$y + 2 = \frac{7 x}{2} - 14$

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

Subtract $2$ from both sides of the equation.

$y = \frac{7 x}{2} - 14 - 2$

Subtract $2$ from $- 14$ .

$y = \frac{7 x}{2} - 16$

Step 5

Reorder terms.

$y = \frac{7}{2} x - 16$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{7}{2} x - 16$

Point-slope form:

$y + 2 = \frac{7}{2} \cdot (x - 4)$

Step 7

Find the Equation Using Point-Slope Formula (4,-2) , (6,5)

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