Algebra Examples

What is an equation of the line that passes through the point

(1, - 3)

$(1, - 3)$ and is perpendicular to the line

x + 3 y = 21

$x + 3 y = 21$

Step 1

Write the problem as a mathematical expression.

(1, - 3)

$(1, - 3)$ ,

x + 3 y = 21

$x + 3 y = 21$

Step 2

Solve

x + 3 y = 21

$x + 3 y = 21$ .

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Step 2.1

Subtract

x

$x$ from both sides of the equation.

3 y = 21 - x

$3 y = 21 - x$

Step 2.2

Divide each term in

3 y = 21 - x

$3 y = 21 - x$ by

3

$3$ and simplify.

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Step 2.2.1

Divide each term in

3 y = 21 - x

$3 y = 21 - x$ by

3

$3$ .

\frac{3 y}{3} = \frac{21}{3} + \frac{- x}{3}

$\frac{3 y}{3} = \frac{21}{3} + \frac{- x}{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of

3

$3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{21}{3} + \frac{- x}{3}$

Step 2.2.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{21}{3} + \frac{- x}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Divide

$21$ by

$3$ .

$y = 7 + \frac{- x}{3}$

Step 2.2.3.1.2

Move the negative in front of the fraction.

$y = 7 - \frac{x}{3}$

Step 3

Find the slope when

$y = 7 - \frac{x}{3}$ .

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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

$7$ and

$- \frac{x}{3}$ .

$y = - \frac{x}{3} + 7$

Step 3.1.3

Write in

$y = m x + b$ form.

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Step 3.1.3.1

Reorder terms.

$y = - (\frac{1}{3} x) + 7$

Step 3.1.3.2

Remove parentheses.

$y = - \frac{1}{3} x + 7$

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

Cancel the common factor of

$1$ and

$- 1$ .

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Step 5.1.1

Rewrite

$1$ as

$- 1 (- 1)$ .

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

Step 5.1.2

Move the negative in front of the fraction.

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

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 3$

Step 5.3

Multiply

$- - (1 \cdot 3)$ .

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Step 5.3.1

Multiply

$3$ by

$1$ .

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

Step 5.3.2

Multiply

$- 1$ by

$3$ .

$m_{perpendicular} = 3$

Step 5.3.3

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

$(1, - 3)$ 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

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

$y - (- 3) = 3 \cdot (x - (1))$

Step 6.2

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

$y + 3 = 3 \cdot (x - 1)$

Step 7

Solve for

$y$ .

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Step 7.1

Simplify

$3 \cdot (x - 1)$ .

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Step 7.1.1

Rewrite.

$y + 3 = 0 + 0 + 3 \cdot (x - 1)$

Step 7.1.2

Simplify by adding zeros.

$y + 3 = 3 \cdot (x - 1)$

Step 7.1.3

Apply the distributive property.

$y + 3 = 3 x + 3 \cdot - 1$

Step 7.1.4

Multiply

$3$ by

$- 1$ .

$y + 3 = 3 x - 3$

Step 7.2

Move all terms not containing

$y$ to the right side of the equation.

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Step 7.2.1

Subtract

$3$ from both sides of the equation.

$y = 3 x - 3 - 3$

Step 7.2.2

Subtract

$3$ from

$- 3$ .

$y = 3 x - 6$

Step 8