Algebra Examples

$(- 5, 2)$ , $y + 3 = 2 x$

Step 1

Subtract $3$ from both sides of the equation.

$y = 2 x - 3$

Step 2

Use the slope-intercept form to find the slope.

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Step 2.1

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 2.2

Using the slope-intercept form, the slope is $2$ .

$m = 2$

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

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 $- \frac{1}{2} \cdot (x + 5)$ .

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Step 5.1.1.1

Rewrite.

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

Step 5.1.1.2

Simplify by adding zeros.

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

Step 5.1.1.3

Apply the distributive property.

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

Step 5.1.1.4

Combine $x$ and $\frac{1}{2}$ .

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

Step 5.1.1.5

Multiply $- \frac{1}{2} \cdot 5$ .

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Step 5.1.1.5.1

Multiply $5$ by $- 1$ .

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

Step 5.1.1.5.2

Combine $- 5$ and $\frac{1}{2}$ .

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

Step 5.1.1.6

Move the negative in front of the fraction.

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

Step 5.1.2

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

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Step 5.1.2.1

Add $2$ to both sides of the equation.

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

Step 5.1.2.2

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

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

Step 5.1.2.3

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

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

Step 5.1.2.4

Combine the numerators over the common denominator.

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

Step 5.1.2.5

Simplify the numerator.

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Step 5.1.2.5.1

Multiply $2$ by $2$ .

$y = - \frac{x}{2} + \frac{- 5 + 4}{2}$

Step 5.1.2.5.2

Add $- 5$ and $4$ .

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

Step 5.1.2.6

Move the negative in front of the fraction.

$y = - \frac{x}{2} - \frac{1}{2}$

Step 5.2

Reorder terms.

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

Step 5.3

Remove parentheses.

$y = - \frac{1}{2} x - \frac{1}{2}$

Step 6