Algebra Examples

Passes through $(6, - 2)$ Perpendicular to $y = - 3 x + 4$

Step 1

Use the slope-intercept form to find the slope.

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Step 1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 1.2

Using the slope-intercept form, the slope is $- 3$ .

$m = - 3$

Step 2

The equation of a perpendicular line must have a slope that is the negative reciprocal of the original slope.

$m_{perpendicular} = - \frac{1}{- 3}$

Step 3

Simplify $- \frac{1}{- 3}$ to find the slope of the perpendicular line.

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Step 3.1

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{3}$

Step 3.2

Multiply $- - \frac{1}{3}$ .

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{1}{3})$

Step 3.2.2

Multiply $\frac{1}{3}$ by $1$ .

$m_{perpendicular} = \frac{1}{3}$

Step 4

Find the equation of the perpendicular line using the point-slope formula.

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Step 4.1

Use the slope $\frac{1}{3}$ and a given point $(6, - 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2) = \frac{1}{3} \cdot (x - (6))$

Step 4.2

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

$y + 2 = \frac{1}{3} \cdot (x - 6)$

Step 5

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

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Step 5.1

Solve for $y$ .

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Step 5.1.1

Simplify $\frac{1}{3} \cdot (x - 6)$ .

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Step 5.1.1.1

Rewrite.

$y + 2 = 0 + 0 + \frac{1}{3} \cdot (x - 6)$

Step 5.1.1.2

Simplify by adding zeros.

$y + 2 = \frac{1}{3} \cdot (x - 6)$

Step 5.1.1.3

Apply the distributive property.

$y + 2 = \frac{1}{3} x + \frac{1}{3} \cdot - 6$

Step 5.1.1.4

Combine $\frac{1}{3}$ and $x$ .

$y + 2 = \frac{x}{3} + \frac{1}{3} \cdot - 6$

Step 5.1.1.5

Cancel the common factor of $3$ .

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Step 5.1.1.5.1

Factor $3$ out of $- 6$ .

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

Step 5.1.1.5.2

Cancel the common factor.

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

Step 5.1.1.5.3

Rewrite the expression.

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

Step 5.1.2

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

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Step 5.1.2.1

Subtract $2$ from both sides of the equation.

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

Step 5.1.2.2

Subtract $2$ from $- 2$ .

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

Step 5.2

Reorder terms.

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

Step 6