Algebra Examples

$y = \frac{2}{3} x + 6$ passing through $(5, - 2)$

Step 1

Find the slope when $y = \frac{2}{3} x + 6$ .

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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 $\frac{2}{3}$ and $x$ .

$y = \frac{2 x}{3} + 6$

Step 1.1.3

Reorder terms.

$y = \frac{2}{3} x + 6$

Step 1.2

Using the slope-intercept form, the slope is $\frac{2}{3}$ .

$m = \frac{2}{3}$

Step 2

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

$m_{perpendicular} = - \frac{1}{\frac{2}{3}}$

Step 3

Simplify $- \frac{1}{\frac{2}{3}}$ to find the slope of the perpendicular line.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{3}{2}))$

Step 3.2

Multiply $\frac{3}{2}$ by $1$ .

$m_{perpendicular} = - \frac{3}{2}$

Step 4

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

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Step 4.1

Use the slope $- \frac{3}{2}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 4.2

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

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

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 $- \frac{3}{2} \cdot (x - 5)$ .

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Step 5.1.1.1

Rewrite.

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

Step 5.1.1.2

Simplify by adding zeros.

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

Step 5.1.1.3

Apply the distributive property.

$y + 2 = - \frac{3}{2} x - \frac{3}{2} \cdot - 5$

Step 5.1.1.4

Combine $x$ and $\frac{3}{2}$ .

$y + 2 = - \frac{x \cdot 3}{2} - \frac{3}{2} \cdot - 5$

Step 5.1.1.5

Multiply $- \frac{3}{2} \cdot - 5$ .

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Step 5.1.1.5.1

Multiply $- 5$ by $- 1$ .

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

Step 5.1.1.5.2

Combine $5$ and $\frac{3}{2}$ .

$y + 2 = - \frac{x \cdot 3}{2} + \frac{5 \cdot 3}{2}$

Step 5.1.1.5.3

Multiply $5$ by $3$ .

$y + 2 = - \frac{x \cdot 3}{2} + \frac{15}{2}$

Step 5.1.1.6

Move $3$ to the left of $x$ .

$y + 2 = - \frac{3 x}{2} + \frac{15}{2}$

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

Subtract $2$ from both sides of the equation.

$y = - \frac{3 x}{2} + \frac{15}{2} - 2$

Step 5.1.2.2

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

$y = - \frac{3 x}{2} + \frac{15}{2} - 2 \cdot \frac{2}{2}$

Step 5.1.2.3

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

$y = - \frac{3 x}{2} + \frac{15}{2} + \frac{- 2 \cdot 2}{2}$

Step 5.1.2.4

Combine the numerators over the common denominator.

$y = - \frac{3 x}{2} + \frac{15 - 2 \cdot 2}{2}$

Step 5.1.2.5

Simplify the numerator.

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Step 5.1.2.5.1

Multiply $- 2$ by $2$ .

$y = - \frac{3 x}{2} + \frac{15 - 4}{2}$

Step 5.1.2.5.2

Subtract $4$ from $15$ .

$y = - \frac{3 x}{2} + \frac{11}{2}$

Step 5.2

Reorder terms.

$y = - (\frac{3}{2} x) + \frac{11}{2}$

Step 5.3

Remove parentheses.

$y = - \frac{3}{2} x + \frac{11}{2}$

Step 6