Find the Equation Using Two Points f(10)=5 and f(2)=-3

$f (10) = 5$ and $f (2) = - 3$

$f (10) = 5$ , which means $(10, 5)$ is a point on the line. $f (2) = - 3$ , which means $(2, - 3)$ is a point on the line, too.

$(10, 5), (2, - 3)$

Find the slope of the line between $(10, 5)$ and $(2, - 3)$ 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.

$m = \frac{- 3 - (5)}{2 - (10)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $5$ .

$m = \frac{- 3 - 5}{2 - (10)}$

Subtract $5$ from $- 3$ .

$m = \frac{- 8}{2 - (10)}$

Simplify the denominator.

Multiply $- 1$ by $10$ .

$m = \frac{- 8}{2 - 10}$

Subtract $10$ from $2$ .

$m = \frac{- 8}{- 8}$

Divide $- 8$ by $- 8$ .

$m = 1$

Use the slope $1$ and a given point $(10, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = 1 \cdot (x - (10))$

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

$y - 5 = 1 \cdot (x - 10)$

Solve for $y$ .

Multiply $x - 10$ by $1$ .

$y - 5 = x - 10$

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

Add $5$ to both sides of the equation.

$y = x - 10 + 5$

Add $- 10$ and $5$ .

$y = x - 5$

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

$f (x) = x - 5$

Find the Equation Using Two Points f(10)=5 and f(2)=-3

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