Find the Equation Using Two Points f(5)=-1 and f(0)=-5

$f (5) = - 1$ and $f (0) = - 5$

$f (5) = - 1$ , which means $(5, - 1)$ is a point on the line. $f (0) = - 5$ , which means $(0, - 5)$ is a point on the line, too.

$(5, - 1), (0, - 5)$

Find the slope of the line between $(5, - 1)$ and $(0, - 5)$ 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.

$m = \frac{- 5 - (- 1)}{0 - (5)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 1$ .

$m = \frac{- 5 + 1}{0 - (5)}$

Add $- 5$ and $1$ .

$m = \frac{- 4}{0 - (5)}$

Simplify the denominator.

Multiply $- 1$ by $5$ .

$m = \frac{- 4}{0 - 5}$

Subtract $5$ from $0$ .

$m = \frac{- 4}{- 5}$

Dividing two negative values results in a positive value.

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

Use the slope $\frac{4}{5}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 1) = \frac{4}{5} \cdot (x - (5))$

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

$y + 1 = \frac{4}{5} \cdot (x - 5)$

Solve for $y$ .

Simplify $\frac{4}{5} \cdot (x - 5)$ .

Rewrite.

$y + 1 = 0 + 0 + \frac{4}{5} \cdot (x - 5)$

Simplify by adding zeros.

$y + 1 = \frac{4}{5} \cdot (x - 5)$

Apply the distributive property.

$y + 1 = \frac{4}{5} x + \frac{4}{5} \cdot - 5$

Combine $\frac{4}{5}$ and $x$ .

$y + 1 = \frac{4 x}{5} + \frac{4}{5} \cdot - 5$

Cancel the common factor of $5$ .

Factor $5$ out of $- 5$ .

$y + 1 = \frac{4 x}{5} + \frac{4}{5} \cdot (5 (- 1))$

Cancel the common factor.

$y + 1 = \frac{4 x}{5} + \frac{4}{5} \cdot (5 \cdot - 1)$

Rewrite the expression.

$y + 1 = \frac{4 x}{5} + 4 \cdot - 1$

Multiply $4$ by $- 1$ .

$y + 1 = \frac{4 x}{5} - 4$

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

Subtract $1$ from both sides of the equation.

$y = \frac{4 x}{5} - 4 - 1$

Subtract $1$ from $- 4$ .

$y = \frac{4 x}{5} - 5$

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

$y = \frac{4}{5} x - 5$

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

$f (x) = \frac{4}{5} x - 5$

Find the Equation Using Two Points f(5)=-1 and f(0)=-5

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