Algebra Examples

$y + 2 = \frac{1}{3} (x - 5)$ , $(- 4, 3)$

Step 1

Solve $y + 2 = \frac{1}{3} \cdot (x - 5)$ .

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Step 1.1

Simplify $\frac{1}{3} (x - 5)$ .

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Step 1.1.1

Rewrite.

$y + 2 = 0 + 0 + \frac{1}{3} (x - 5)$

Step 1.1.2

Simplify by adding zeros.

$y + 2 = \frac{1}{3} (x - 5)$

Step 1.1.3

Apply the distributive property.

$y + 2 = \frac{1}{3} x + \frac{1}{3} \cdot - 5$

Step 1.1.4

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

$y + 2 = \frac{x}{3} + \frac{1}{3} \cdot - 5$

Step 1.1.5

Combine $\frac{1}{3}$ and $- 5$ .

$y + 2 = \frac{x}{3} + \frac{- 5}{3}$

Step 1.1.6

Move the negative in front of the fraction.

$y + 2 = \frac{x}{3} - \frac{5}{3}$

Step 1.2

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

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Step 1.2.1

Subtract $2$ from both sides of the equation.

$y = \frac{x}{3} - \frac{5}{3} - 2$

Step 1.2.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{x}{3} - \frac{5}{3} - 2 \cdot \frac{3}{3}$

Step 1.2.3

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

$y = \frac{x}{3} - \frac{5}{3} + \frac{- 2 \cdot 3}{3}$

Step 1.2.4

Combine the numerators over the common denominator.

$y = \frac{x}{3} + \frac{- 5 - 2 \cdot 3}{3}$

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $- 2$ by $3$ .

$y = \frac{x}{3} + \frac{- 5 - 6}{3}$

Step 1.2.5.2

Subtract $6$ from $- 5$ .

$y = \frac{x}{3} + \frac{- 11}{3}$

Step 1.2.6

Move the negative in front of the fraction.

$y = \frac{x}{3} - \frac{11}{3}$

Step 2

Find the slope when $y = \frac{x}{3} - \frac{11}{3}$ .

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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

Reorder terms.

$y = \frac{1}{3} x - \frac{11}{3}$

Step 2.2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Multiply $- (1 \cdot 3)$ .

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Step 4.2.1

Multiply $3$ by $1$ .

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

Step 4.2.2

Multiply $- 1$ by $3$ .

$m_{perpendicular} = - 3$

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Solve for $y$ .

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Step 6.1

Simplify $- 3 \cdot (x + 4)$ .

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Step 6.1.1

Rewrite.

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

Step 6.1.2

Simplify by adding zeros.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.1.4

Multiply $- 3$ by $4$ .

$y - 3 = - 3 x - 12$

Step 6.2

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

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Step 6.2.1

Add $3$ to both sides of the equation.

$y = - 3 x - 12 + 3$

Step 6.2.2

Add $- 12$ and $3$ .

$y = - 3 x - 9$

Step 7