Algebra Examples

A line is perpendicular to $y = \frac{2}{3} x + \frac{2}{3}$ and intersects the point $(- 2, 4)$ What is the equation of this perpendicular line?

Step 1

Write the problem as a mathematical expression.

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

Step 2

Find the slope when $y = \frac{2}{3} x + \frac{2}{3}$ .

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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

Simplify the right side.

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Step 2.1.2.1

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

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

Step 2.1.3

Reorder terms.

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

Step 2.2

Using the slope-intercept form, the slope is $\frac{2}{3}$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Multiply $\frac{3}{2}$ by $1$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

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

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Step 6.1.1.1

Rewrite.

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

Step 6.1.1.2

Simplify terms.

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Step 6.1.1.2.1

Apply the distributive property.

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

Step 6.1.1.2.2

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

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

Step 6.1.1.2.3

Cancel the common factor of $2$ .

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Step 6.1.1.2.3.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Step 6.1.1.2.3.2

Cancel the common factor.

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

Step 6.1.1.2.3.3

Rewrite the expression.

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

Step 6.1.1.3

Move $3$ to the left of $x$ .

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

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 $4$ to both sides of the equation.

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

Step 6.1.2.2

Add $- 3$ and $4$ .

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

Step 6.2

Reorder terms.

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

Step 6.3

Remove parentheses.

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

Step 7