Algebra Examples

$(3, 5)$ , $m = 5$

Step 1

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

$m_{perpendicular} = - \frac{1}{5}$

Step 2

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

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Step 2.1

Use the slope $- \frac{1}{5}$ and a given point $(3, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 2.2

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

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

Step 3

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

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Step 3.1

Solve for $y$ .

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Step 3.1.1

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

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Step 3.1.1.1

Rewrite.

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

Step 3.1.1.2

Simplify by adding zeros.

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

Step 3.1.1.3

Apply the distributive property.

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

Step 3.1.1.4

Combine $x$ and $\frac{1}{5}$ .

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

Step 3.1.1.5

Multiply $- \frac{1}{5} \cdot - 3$ .

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Step 3.1.1.5.1

Multiply $- 3$ by $- 1$ .

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

Step 3.1.1.5.2

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

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

Step 3.1.2

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

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Step 3.1.2.1

Add $5$ to both sides of the equation.

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

Step 3.1.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 3.1.2.3

Combine $5$ and $\frac{5}{5}$ .

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

Step 3.1.2.4

Combine the numerators over the common denominator.

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

Step 3.1.2.5

Simplify the numerator.

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Step 3.1.2.5.1

Multiply $5$ by $5$ .

$y = - \frac{x}{5} + \frac{3 + 25}{5}$

Step 3.1.2.5.2

Add $3$ and $25$ .

$y = - \frac{x}{5} + \frac{28}{5}$

Step 3.2

Reorder terms.

$y = - (\frac{1}{5} x) + \frac{28}{5}$

Step 3.3

Remove parentheses.

$y = - \frac{1}{5} x + \frac{28}{5}$

Step 4