Algebra Examples

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

Step 1

Solve $y - 1 = - \frac{1}{3} \cdot (x - 4)$ .

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Step 1.1

Simplify $- \frac{1}{3} (x - 4)$ .

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Step 1.1.1

Rewrite.

$y - 1 = 0 + 0 - \frac{1}{3} (x - 4)$

Step 1.1.2

Simplify by adding zeros.

$y - 1 = - \frac{1}{3} (x - 4)$

Step 1.1.3

Apply the distributive property.

$y - 1 = - \frac{1}{3} x - \frac{1}{3} \cdot - 4$

Step 1.1.4

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

$y - 1 = - \frac{x}{3} - \frac{1}{3} \cdot - 4$

Step 1.1.5

Multiply $- \frac{1}{3} \cdot - 4$ .

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Step 1.1.5.1

Multiply $- 4$ by $- 1$ .

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

Step 1.1.5.2

Combine $4$ and $\frac{1}{3}$ .

$y - 1 = - \frac{x}{3} + \frac{4}{3}$

Step 1.2

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

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Step 1.2.1

Add $1$ to both sides of the equation.

$y = - \frac{x}{3} + \frac{4}{3} + 1$

Step 1.2.2

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

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.2.4

Add $4$ and $3$ .

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

Step 2

Find the slope when $y = - \frac{x}{3} + \frac{7}{3}$ .

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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

Write in $y = m x + b$ form.

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Step 2.1.2.1

Reorder terms.

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

Step 2.1.2.2

Remove parentheses.

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

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

Cancel the common factor of $1$ and $- 1$ .

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Step 4.1.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 4.1.2

Move the negative in front of the fraction.

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 3$

Step 4.3

Multiply $- - (1 \cdot 3)$ .

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Step 4.3.1

Multiply $3$ by $1$ .

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

Step 4.3.2

Multiply $- 1$ by $3$ .

$m_{perpendicular} = 3$

Step 4.3.3

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 $(- 2, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = 3 \cdot (x - (- 2))$

Step 5.2

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

$y - 5 = 3 \cdot (x + 2)$

Step 6

Solve for $y$ .

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Step 6.1

Simplify $3 \cdot (x + 2)$ .

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Step 6.1.1

Rewrite.

$y - 5 = 0 + 0 + 3 \cdot (x + 2)$

Step 6.1.2

Simplify by adding zeros.

$y - 5 = 3 \cdot (x + 2)$

Step 6.1.3

Apply the distributive property.

$y - 5 = 3 x + 3 \cdot 2$

Step 6.1.4

Multiply $3$ by $2$ .

$y - 5 = 3 x + 6$

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 $5$ to both sides of the equation.

$y = 3 x + 6 + 5$

Step 6.2.2

Add $6$ and $5$ .

$y = 3 x + 11$

Step 7