Algebra Examples

$f (- 2) = 6$ and $f (0) = - 4$

Step 1

$f (- 2) = 6$ , which means $(- 2, 6)$ is a point on the line. $f (0) = - 4$ , which means $(0, - 4)$ is a point on the line, too.

$(- 2, 6), (0, - 4)$

Step 2

Find the slope of the line between $(- 2, 6)$ and $(0, - 4)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

Tap for more steps...

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.

$m = \frac{- 4 - (6)}{0 - (- 2)}$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 2.4.1.1

Cancel the common factor of $- 4 - (6)$ and $0 - (- 2)$ .

Tap for more steps...

Step 2.4.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

$m = \frac{- 1 \cdot 4 - (6)}{0 - (- 2)}$

Step 2.4.1.1.2

Factor $- 1$ out of $- 1 (4) - (6)$ .

$m = \frac{- 1 (4 + 6)}{0 - (- 2)}$

Step 2.4.1.1.3

Reorder terms.

$m = \frac{- 1 (4 + 6)}{0 - 2 \cdot - 1}$

Step 2.4.1.1.4

Factor $2$ out of $- 1 (4 + 6)$ .

$m = \frac{2 (- 1 (2 + 3))}{0 - 2 \cdot - 1}$

Step 2.4.1.1.5

Cancel the common factors.

Tap for more steps...

Step 2.4.1.1.5.1

Factor $2$ out of $0$ .

$m = \frac{2 (- 1 (2 + 3))}{2 (0) - 2 \cdot - 1}$

Step 2.4.1.1.5.2

Factor $2$ out of $- 2 \cdot - 1$ .

$m = \frac{2 (- 1 (2 + 3))}{2 (0) + 2 (1)}$

Step 2.4.1.1.5.3

Factor $2$ out of $2 (0) + 2 (- - 1)$ .

$m = \frac{2 (- 1 (2 + 3))}{2 (0 + 1)}$

Step 2.4.1.1.5.4

Cancel the common factor.

$m = \frac{2 (- 1 (2 + 3))}{2 (0 + 1)}$

Step 2.4.1.1.5.5

Rewrite the expression.

$m = \frac{- 1 (2 + 3)}{0 + 1}$

Step 2.4.1.2

Add $2$ and $3$ .

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

Step 2.4.2

Simplify the denominator.

Tap for more steps...

Step 2.4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.4.2.2

Add $0$ and $1$ .

$m = \frac{- 1 \cdot 5}{1}$

Step 2.4.3

Simplify the expression.

Tap for more steps...

Step 2.4.3.1

Multiply $- 1$ by $5$ .

$m = \frac{- 5}{1}$

Step 2.4.3.2

Divide $- 5$ by $1$ .

$m = - 5$

Step 3

Use the slope $- 5$ and a given point $(- 2, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = - 5 \cdot (x - (- 2))$

Step 4

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

$y - 6 = - 5 \cdot (x + 2)$

Step 5

Solve for $y$ .

Tap for more steps...

Step 5.1

Simplify $- 5 \cdot (x + 2)$ .

Tap for more steps...

Step 5.1.1

Rewrite.

$y - 6 = 0 + 0 - 5 \cdot (x + 2)$

Step 5.1.2

Simplify by adding zeros.

$y - 6 = - 5 \cdot (x + 2)$

Step 5.1.3

Apply the distributive property.

$y - 6 = - 5 x - 5 \cdot 2$

Step 5.1.4

Multiply $- 5$ by $2$ .

$y - 6 = - 5 x - 10$

Step 5.2

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

Tap for more steps...

Step 5.2.1

Add $6$ to both sides of the equation.

$y = - 5 x - 10 + 6$

Step 5.2.2

Add $- 10$ and $6$ .

$y = - 5 x - 4$

Step 6

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

$f (x) = - 5 x - 4$

Step 7