Algebra Examples

$y = \frac{2}{3} x$ and passes through the point

$(3, 2)$

Step 1

Find the slope when

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

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Step 1.1

Rewrite in slope-intercept form.

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Step 1.1.1

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 1.1.2

Simplify the right side.

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Step 1.1.2.1

Combine

$\frac{2}{3}$ and

$x$ .

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

Step 1.1.3

Reorder terms.

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

Step 1.2

Using the slope-intercept form, the slope is

$\frac{2}{3}$ .

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

Step 2

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

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

Step 3

Simplify

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

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Multiply

$\frac{3}{2}$ by

$1$ .

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

Step 4

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

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Step 4.1

Use the slope

$- \frac{3}{2}$ 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

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

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

Step 4.2

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

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

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

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

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Step 5.1.1.1

Rewrite.

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

Step 5.1.1.2

Simplify by adding zeros.

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

Step 5.1.1.3

Apply the distributive property.

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

Step 5.1.1.4

Combine

$x$ and

$\frac{3}{2}$ .

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

Step 5.1.1.5

Multiply

$- \frac{3}{2} \cdot - 3$ .

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Step 5.1.1.5.1

Multiply

$- 3$ by

$- 1$ .

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

Step 5.1.1.5.2

Combine

$3$ and

$\frac{3}{2}$ .

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

Step 5.1.1.5.3

Multiply

$3$ by

$3$ .

$y - 2 = - \frac{x \cdot 3}{2} + \frac{9}{2}$

Step 5.1.1.6

Move

$3$ to the left of

$x$ .

$y - 2 = - \frac{3 x}{2} + \frac{9}{2}$

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

Add

$2$ to both sides of the equation.

$y = - \frac{3 x}{2} + \frac{9}{2} + 2$

Step 5.1.2.2

To write

$2$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$y = - \frac{3 x}{2} + \frac{9}{2} + 2 \cdot \frac{2}{2}$

Step 5.1.2.3

Combine

$2$ and

$\frac{2}{2}$ .

$y = - \frac{3 x}{2} + \frac{9}{2} + \frac{2 \cdot 2}{2}$

Step 5.1.2.4

Combine the numerators over the common denominator.

$y = - \frac{3 x}{2} + \frac{9 + 2 \cdot 2}{2}$

Step 5.1.2.5

Simplify the numerator.

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Step 5.1.2.5.1

Multiply

$2$ by

$2$ .

$y = - \frac{3 x}{2} + \frac{9 + 4}{2}$

Step 5.1.2.5.2

Add

$9$ and

$4$ .

$y = - \frac{3 x}{2} + \frac{13}{2}$

Step 5.2

Reorder terms.

$y = - (\frac{3}{2} x) + \frac{13}{2}$

Step 5.3

Remove parentheses.

$y = - \frac{3}{2} x + \frac{13}{2}$

Step 6