Find the Equation Using Two Points f(-5)=-1 and f(2)=4

$f (- 5) = - 1$ and $f (2) = 4$

$f (- 5) = - 1$ , which means $(- 5, - 1)$ is a point on the line. $f (2) = 4$ , which means $(2, 4)$ is a point on the line, too.

$(- 5, - 1), (2, 4)$

Find the slope of the line between $(- 5, - 1)$ and $(2, 4)$ 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.

$m = \frac{4 - (- 1)}{2 - (- 5)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 1$ .

$m = \frac{4 + 1}{2 - (- 5)}$

Add $4$ and $1$ .

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

Simplify the denominator.

Multiply $- 1$ by $- 5$ .

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

Add $2$ and $5$ .

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

Use the slope $\frac{5}{7}$ and a given point $(- 5, - 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 1) = \frac{5}{7} \cdot (x - (- 5))$

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

$y + 1 = \frac{5}{7} \cdot (x + 5)$

Solve for $y$ .

Simplify $\frac{5}{7} \cdot (x + 5)$ .

Rewrite.

$y + 1 = 0 + 0 + \frac{5}{7} \cdot (x + 5)$

Simplify by adding zeros.

$y + 1 = \frac{5}{7} \cdot (x + 5)$

Apply the distributive property.

$y + 1 = \frac{5}{7} x + \frac{5}{7} \cdot 5$

Combine $\frac{5}{7}$ and $x$ .

$y + 1 = \frac{5 x}{7} + \frac{5}{7} \cdot 5$

Multiply $\frac{5}{7} \cdot 5$ .

Combine $\frac{5}{7}$ and $5$ .

$y + 1 = \frac{5 x}{7} + \frac{5 \cdot 5}{7}$

Multiply $5$ by $5$ .

$y + 1 = \frac{5 x}{7} + \frac{25}{7}$

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

Subtract $1$ from both sides of the equation.

$y = \frac{5 x}{7} + \frac{25}{7} - 1$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = \frac{5 x}{7} + \frac{25}{7} - 1 \cdot \frac{7}{7}$

Combine $- 1$ and $\frac{7}{7}$ .

$y = \frac{5 x}{7} + \frac{25}{7} + \frac{- 1 \cdot 7}{7}$

Combine the numerators over the common denominator.

$y = \frac{5 x}{7} + \frac{25 - 1 \cdot 7}{7}$

Simplify the numerator.

Multiply $- 1$ by $7$ .

$y = \frac{5 x}{7} + \frac{25 - 7}{7}$

Subtract $7$ from $25$ .

$y = \frac{5 x}{7} + \frac{18}{7}$

The final answer is the equation in slope-intercept form.

$y = \frac{5}{7} x + \frac{18}{7}$

Replace $y$ by $f (x)$ .

$f (x) = \frac{5}{7} x + \frac{18}{7}$

Find the Equation Using Two Points f(-5)=-1 and f(2)=4

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