Find the Equation Using Two Points f(5)=7 and f(-2)=0

$f (5) = 7$ and $f (- 2) = 0$

Step 1

$f (5) = 7$ , which means $(5, 7)$ is a point on the line. $f (- 2) = 0$ , which means $(- 2, 0)$ is a point on the line, too.

$(5, 7), (- 2, 0)$

Step 2

Find the slope of the line between $(5, 7)$ and $(- 2, 0)$ 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.

$m = \frac{0 - (7)}{- 2 - (5)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $7$ .

$m = \frac{0 - 7}{- 2 - (5)}$

Subtract $7$ from $0$ .

$m = \frac{- 7}{- 2 - (5)}$

Simplify the denominator.

Multiply $- 1$ by $5$ .

$m = \frac{- 7}{- 2 - 5}$

Subtract $5$ from $- 2$ .

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

Divide $- 7$ by $- 7$ .

$m = 1$

Step 3

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

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

Step 4

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

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

Step 5

Solve for $y$ .

Multiply $x - 5$ by $1$ .

$y - 7 = x - 5$

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

Add $7$ to both sides of the equation.

$y = x - 5 + 7$

Add $- 5$ and $7$ .

$y = x + 2$

Step 6

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

$f (x) = x + 2$

Step 7

Find the Equation Using Two Points f(5)=7 and f(-2)=0

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