Finite Math Examples

$f (10) = 0$ , $f (20) = 10$

Step 1

$f (10) = 0$ , which means $(10, 0)$ is a point on the line. $f (20) = 10$ , which means $(20, 10)$ is a point on the line, too.

$(10, 0), (20, 10)$

Step 2

Find the slope of the line between $(10, 0)$ and $(20, 10)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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Step 2.1

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 2.2

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).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 2.3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{10 - (0)}{20 - (10)}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Multiply $- 1$ by $0$ .

$m = \frac{10 + 0}{20 - (10)}$

Step 2.4.1.2

Add $10$ and $0$ .

$m = \frac{10}{20 - (10)}$

Step 2.4.2

Simplify the denominator.

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Step 2.4.2.1

Multiply $- 1$ by $10$ .

$m = \frac{10}{20 - 10}$

Step 2.4.2.2

Subtract $10$ from $20$ .

$m = \frac{10}{10}$

Step 2.4.3

Divide $10$ by $10$ .

$m = 1$

Step 3

Use the slope $1$ and a given point $(10, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = 1 \cdot (x - (10))$

Step 4

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

$y + 0 = 1 \cdot (x - 10)$

Step 5

Solve for $y$ .

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Step 5.1

Add $y$ and $0$ .

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

Step 5.2

Multiply $x - 10$ by $1$ .

$y = x - 10$

Step 6

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

$f (x) = x - 10$

Step 7