Algebra Examples

Passing through $(3, - 5)$ and perpendicular to the line whose equation is $y = \frac{1}{5} x + 2$

Step 1

Find the slope when $y = \frac{1}{5} x + 2$ .

Tap for more steps...

Step 1.1

Rewrite in slope-intercept form.

Tap for more steps...

Step 1.1.1

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 1.1.2

Simplify the right side.

Tap for more steps...

Step 1.1.2.1

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

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

Step 1.1.3

Reorder terms.

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

Step 1.2

Using the slope-intercept form, the slope is $\frac{1}{5}$ .

$m = \frac{1}{5}$

Step 2

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

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

Step 3

Simplify $- \frac{1}{\frac{1}{5}}$ to find the slope of the perpendicular line.

Tap for more steps...

Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 5)$

Step 3.2

Multiply $- (1 \cdot 5)$ .

Tap for more steps...

Step 3.2.1

Multiply $5$ by $1$ .

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

Step 3.2.2

Multiply $- 1$ by $5$ .

$m_{perpendicular} = - 5$

Step 4

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

Tap for more steps...

Step 4.1

Use the slope $- 5$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 4.2

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

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

Step 5

Solve for $y$ .

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Rewrite.

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

Step 5.1.2

Simplify by adding zeros.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.1.4

Multiply $- 5$ by $- 3$ .

$y + 5 = - 5 x + 15$

Step 5.2

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

Tap for more steps...

Step 5.2.1

Subtract $5$ from both sides of the equation.

$y = - 5 x + 15 - 5$

Step 5.2.2

Subtract $5$ from $15$ .

$y = - 5 x + 10$

Step 6