Algebra Examples

What is the equation of the line that is perpendicular to the line defined by the equation

2 y = 3 x + 8

$2 y = 3 x + 8$ and goes through the point

(3, 2)

$(3, 2)$ ?

Step 1

Divide each term in

$2 y = 3 x + 8$ by

$2$ and simplify.

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Step 1.1

Divide each term in

$2 y = 3 x + 8$ by

$2$ .

$\frac{2 y}{2} = \frac{3 x}{2} + \frac{8}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of

$2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{3 x}{2} + \frac{8}{2}$

Step 1.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{3 x}{2} + \frac{8}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide

$8$ by

$2$ .

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

Step 2

Find the slope when

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

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Step 2.1

Rewrite in slope-intercept form.

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Step 2.1.1

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 2.1.2

Reorder terms.

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

Step 2.2

Using the slope-intercept form, the slope is

$\frac{3}{2}$ .

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

Step 3

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

$m_{perpendicular} = - \frac{1}{\frac{3}{2}}$

Step 4

Simplify

$- \frac{1}{\frac{3}{2}}$ to find the slope of the perpendicular line.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{2}{3}))$

Step 4.2

Multiply

$\frac{2}{3}$ by

$1$ .

$m_{perpendicular} = - \frac{2}{3}$

Step 5

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

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Step 5.1

Use the slope

$- \frac{2}{3}$ and a given point

$(3, 2)$ 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

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

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

Step 5.2

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

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

Step 6

Write in

$y = m x + b$ form.

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Step 6.1

Solve for

$y$ .

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Step 6.1.1

Simplify

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

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Step 6.1.1.1

Rewrite.

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

Step 6.1.1.2

Simplify terms.

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Step 6.1.1.2.1

Apply the distributive property.

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

Step 6.1.1.2.2

Combine

$x$ and

$\frac{2}{3}$ .

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

Step 6.1.1.2.3

Cancel the common factor of

$3$ .

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Step 6.1.1.2.3.1

Move the leading negative in

$- \frac{2}{3}$ into the numerator.

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

Step 6.1.1.2.3.2

Factor

$3$ out of

$- 3$ .

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

Step 6.1.1.2.3.3

Cancel the common factor.

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

Step 6.1.1.2.3.4

Rewrite the expression.

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

Step 6.1.1.2.4

Multiply

$- 2$ by

$- 1$ .

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

Step 6.1.1.3

Move

$2$ to the left of

$x$ .

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

Step 6.1.2

Move all terms not containing

$y$ to the right side of the equation.

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Step 6.1.2.1

Add

$2$ to both sides of the equation.

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

Step 6.1.2.2

Add

$2$ and

$2$ .

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

Step 6.2

Reorder terms.

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

Step 6.3

Remove parentheses.

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

Step 7