Algebra Examples

Passing through

(5, - 4)

$(5, - 4)$ and perpendicular to the line whose equation is

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

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

Step 1

Find the slope when

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

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

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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

$y = m x + b$ , where

m

$m$ is the slope and

b

$b$ is the y-intercept.

y = m x + b

$y = m x + b$

Step 1.1.2

Simplify the right side.

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Step 1.1.2.1

Combine

\frac{1}{3}

$\frac{1}{3}$ and

x

$x$ .

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

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

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

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

Step 1.1.3

Reorder terms.

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

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

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

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

Step 1.2

Using the slope-intercept form, the slope is

\frac{1}{3}

$\frac{1}{3}$ .

m = \frac{1}{3}

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

m = \frac{1}{3}

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

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{1}{3}}

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

Step 3

Simplify

- \frac{1}{\frac{1}{3}}

$- \frac{1}{\frac{1}{3}}$ 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 \cdot 3)

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

Step 3.2

Multiply

- (1 \cdot 3)

$- (1 \cdot 3)$ .

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Step 3.2.1

Multiply

3

$3$ by

1

$1$ .

m_{perpendicular} = - 1 \cdot 3

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

Step 3.2.2

Multiply

- 1

$- 1$ by

3

$3$ .

m_{perpendicular} = - 3

$m_{perpendicular} = - 3$

m_{perpendicular} = - 3

$m_{perpendicular} = - 3$

m_{perpendicular} = - 3

$m_{perpendicular} = - 3$

Step 4

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

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Step 4.1

Use the slope

- 3

$- 3$ and a given point

(5, - 4)

$(5, - 4)$ to substitute for

x_{1}

$x_{1}$ and

y_{1}

$y_{1}$ in the point-slope form

y - y_{1} = m (x - x_{1})

$y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation

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

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

y - (- 4) = - 3 \cdot (x - (5))

$y - (- 4) = - 3 \cdot (x - (5))$

Step 4.2

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

y + 4 = - 3 \cdot (x - 5)

$y + 4 = - 3 \cdot (x - 5)$

y + 4 = - 3 \cdot (x - 5)

$y + 4 = - 3 \cdot (x - 5)$

Step 5

Solve for

y

$y$ .

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Step 5.1

Simplify

- 3 \cdot (x - 5)

$- 3 \cdot (x - 5)$ .

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Step 5.1.1

Rewrite.

y + 4 = 0 + 0 - 3 \cdot (x - 5)

$y + 4 = 0 + 0 - 3 \cdot (x - 5)$

Step 5.1.2

Simplify by adding zeros.

$y + 4 = - 3 \cdot (x - 5)$

Step 5.1.3

Apply the distributive property.

$y + 4 = - 3 x - 3 \cdot - 5$

Step 5.1.4

Multiply

$- 3$ by

$- 5$ .

$y + 4 = - 3 x + 15$

Step 5.2

Move all terms not containing

$y$ to the right side of the equation.

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Step 5.2.1

Subtract

$4$ from both sides of the equation.

$y = - 3 x + 15 - 4$

Step 5.2.2

Subtract

$4$ from

$15$ .

$y = - 3 x + 11$

Step 6