Algebra Examples

$(a, b)$ , $(c, d)$

Step 1

Find the slope of the line between $(a, b)$ and $(c, d)$ 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 1.1

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 1.2

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 1.3

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

$m = \frac{d - (b)}{c - (a)}$

Step 1.4

Multiply $- 1$ by $a$ .

$m = \frac{d - b}{c - a}$

Step 2

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

$y - (b) = \frac{d - b}{c - a} \cdot (x - (a))$

Step 3

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

$y - b = \frac{d - b}{c - a} \cdot (x - a)$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $\frac{d - b}{c - a} \cdot (x - a)$ .

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Step 4.1.1

Rewrite.

$y - b = 0 + 0 + \frac{d - b}{c - a} \cdot (x - a)$

Step 4.1.2

Simplify by adding zeros.

$y - b = \frac{d - b}{c - a} \cdot (x - a)$

Step 4.1.3

Multiply $\frac{d - b}{c - a}$ by $x - a$ .

$y - b = \frac{(d - b) (x - a)}{c - a}$

Step 4.2

Add $b$ to both sides of the equation.

$y = \frac{(d - b) (x - a)}{c - a} + b$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = \frac{(d - b) (x - a)}{c - a} + b$

Point-slope form:

$y - b = \frac{d - b}{c - a} \cdot (x - a)$

Step 6