Algebra Examples

the line through $(0, 2)$ , perpendicular to $y = \frac{5}{2} x + 2$

Step 1

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

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Step 1.1

Rewrite in slope-intercept form.

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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.

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Step 1.1.2.1

Combine $\frac{5}{2}$ and $x$ .

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

Step 1.1.3

Reorder terms.

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

Step 1.2

Using the slope-intercept form, the slope is $\frac{5}{2}$ .

$m = \frac{5}{2}$

Step 2

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

$m_{perpendicular} = - \frac{1}{\frac{5}{2}}$

Step 3

Simplify $- \frac{1}{\frac{5}{2}}$ to find the slope of the perpendicular line.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{2}{5}))$

Step 3.2

Multiply $\frac{2}{5}$ by $1$ .

$m_{perpendicular} = - \frac{2}{5}$

Step 4

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

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Step 4.1

Use the slope $- \frac{2}{5}$ and a given point $(0, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - \frac{2}{5} \cdot (x - (0))$

Step 4.2

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

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

Step 5

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

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Step 5.1

Solve for $y$ .

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Step 5.1.1

Simplify $- \frac{2}{5} \cdot (x + 0)$ .

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Step 5.1.1.1

Add $x$ and $0$ .

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

Step 5.1.1.2

Combine $x$ and $\frac{2}{5}$ .

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

Step 5.1.1.3

Move $2$ to the left of $x$ .

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

Step 5.1.2

Add $2$ to both sides of the equation.

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

Step 5.2

Reorder terms.

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

Step 5.3

Remove parentheses.

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

Step 6