Algebra Examples

$(3, 4)$ and $(- 6, - 2)$

Step 1

Find the slope of the line between $(3, 4)$ and $(- 6, - 2)$ 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.

$m = \frac{- 2 - (4)}{- 6 - (3)}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Multiply $- 1$ by $4$ .

$m = \frac{- 2 - 4}{- 6 - (3)}$

Step 1.4.1.2

Subtract $4$ from $- 2$ .

$m = \frac{- 6}{- 6 - (3)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $3$ .

$m = \frac{- 6}{- 6 - 3}$

Step 1.4.2.2

Subtract $3$ from $- 6$ .

$m = \frac{- 6}{- 9}$

Step 1.4.3

Cancel the common factor of $- 6$ and $- 9$ .

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Step 1.4.3.1

Factor $- 3$ out of $- 6$ .

$m = \frac{- 3 \cdot 2}{- 9}$

Step 1.4.3.2

Cancel the common factors.

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Step 1.4.3.2.1

Factor $- 3$ out of $- 9$ .

$m = \frac{- 3 \cdot 2}{- 3 \cdot 3}$

Step 1.4.3.2.2

Cancel the common factor.

$m = \frac{- 3 \cdot 2}{- 3 \cdot 3}$

Step 1.4.3.2.3

Rewrite the expression.

$m = \frac{2}{3}$

Step 2

Use the slope $\frac{2}{3}$ and a given point $(3, 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = \frac{2}{3} \cdot (x - (3))$

Step 3

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

$y - 4 = \frac{2}{3} \cdot (x - 3)$

Step 4

Write the equation in $y = m x + b$ form.

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Step 4.1

Solve for $y$ .

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Step 4.1.1

Simplify $\frac{2}{3} \cdot (x - 3)$ .

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Step 4.1.1.1

Rewrite.

$y - 4 = 0 + 0 + \frac{2}{3} \cdot (x - 3)$

Step 4.1.1.2

Simplify by adding zeros.

$y - 4 = \frac{2}{3} \cdot (x - 3)$

Step 4.1.1.3

Apply the distributive property.

$y - 4 = \frac{2}{3} x + \frac{2}{3} \cdot - 3$

Step 4.1.1.4

Combine $\frac{2}{3}$ and $x$ .

$y - 4 = \frac{2 x}{3} + \frac{2}{3} \cdot - 3$

Step 4.1.1.5

Cancel the common factor of $3$ .

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Step 4.1.1.5.1

Factor $3$ out of $- 3$ .

$y - 4 = \frac{2 x}{3} + \frac{2}{3} \cdot (3 (- 1))$

Step 4.1.1.5.2

Cancel the common factor.

$y - 4 = \frac{2 x}{3} + \frac{2}{3} \cdot (3 \cdot - 1)$

Step 4.1.1.5.3

Rewrite the expression.

$y - 4 = \frac{2 x}{3} + 2 \cdot - 1$

Step 4.1.1.6

Multiply $2$ by $- 1$ .

$y - 4 = \frac{2 x}{3} - 2$

Step 4.1.2

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

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Step 4.1.2.1

Add $4$ to both sides of the equation.

$y = \frac{2 x}{3} - 2 + 4$

Step 4.1.2.2

Add $- 2$ and $4$ .

$y = \frac{2 x}{3} + 2$

Step 4.2

Reorder terms.

$y = \frac{2}{3} x + 2$

Step 5