Algebra Examples

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

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

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

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Step 2.1

Simplify $\frac{1}{3} \cdot (x + 2)$ .

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Step 2.1.1

Rewrite.

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

Step 2.1.2

Simplify by adding zeros.

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

Step 2.1.3

Apply the distributive property.

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

Step 2.1.4

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

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

Step 2.1.5

Combine $\frac{1}{3}$ and $2$ .

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

Step 2.2

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

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Step 2.2.1

Add $1$ to both sides of the equation.

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

Step 2.2.2

Write $1$ as a fraction with a common denominator.

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

Step 2.2.3

Combine the numerators over the common denominator.

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

Step 2.2.4

Add $2$ and $3$ .

$y = \frac{x}{3} + \frac{5}{3}$

Step 3

Find the slope when $y = \frac{x}{3} + \frac{5}{3}$ .

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Step 3.1

Rewrite in slope-intercept form.

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Step 3.1.1

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 3.1.2

Reorder terms.

$y = \frac{1}{3} x + \frac{5}{3}$

Step 3.2

Using the slope-intercept form, the slope is $\frac{1}{3}$ .

$m = \frac{1}{3}$

Step 4

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

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

Step 5

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

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Step 5.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2

Multiply $- (1 \cdot 3)$ .

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Step 5.2.1

Multiply $3$ by $1$ .

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

Step 5.2.2

Multiply $- 1$ by $3$ .

$m_{perpendicular} = - 3$

Step 6

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

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Step 6.1

Use the slope $- 3$ and a given point $(0, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = - 3 \cdot (x - (0))$

Step 6.2

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

$y + 0 = - 3 \cdot (x + 0)$

Step 7

Solve for $y$ .

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Step 7.1

Add $y$ and $0$ .

$y = - 3 \cdot (x + 0)$

Step 7.2

Add $x$ and $0$ .

$y = - 3 x$

Step 8