Algebra Examples

line

y = \frac{2}{3} x - 4

$y = \frac{2}{3} x - 4$ , point

(2, - 6)

$(2, - 6)$

Step 1

Find the slope when

y = \frac{2}{3} x - 4

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

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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

$y = m x + b$ , where

m

$m$ is the slope and

b

$b$ is the y-intercept.

y = m x + b

$y = m x + b$

Step 1.1.2

Simplify the right side.

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Step 1.1.2.1

Combine

\frac{2}{3}

$\frac{2}{3}$ and

x

$x$ .

y = \frac{2 x}{3} - 4

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

y = \frac{2 x}{3} - 4

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

Step 1.1.3

Reorder terms.

y = \frac{2}{3} x - 4

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

y = \frac{2}{3} x - 4

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

Step 1.2

Using the slope-intercept form, the slope is

\frac{2}{3}

$\frac{2}{3}$ .

m = \frac{2}{3}

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

m = \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}}

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

Step 3

Simplify

- \frac{1}{\frac{2}{3}}

$- \frac{1}{\frac{2}{3}}$ 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}))

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

Step 3.2

Multiply

\frac{3}{2}

$\frac{3}{2}$ by

1

$1$ .

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

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

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

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

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}

$- \frac{3}{2}$ and a given point

(2, - 6)

$(2, - 6)$ to substitute for

x_{1}

$x_{1}$ and

y_{1}

$y_{1}$ in the point-slope form

y - y_{1} = m (x - x_{1})

$y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

y - (- 6) = - \frac{3}{2} \cdot (x - (2))

$y - (- 6) = - \frac{3}{2} \cdot (x - (2))$

Step 4.2

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

y + 6 = - \frac{3}{2} \cdot (x - 2)

$y + 6 = - \frac{3}{2} \cdot (x - 2)$

y + 6 = - \frac{3}{2} \cdot (x - 2)

$y + 6 = - \frac{3}{2} \cdot (x - 2)$

Step 5

Write in

y = m x + b

$y = m x + b$ form.

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Step 5.1

Solve for

y

$y$ .

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Step 5.1.1

Simplify

- \frac{3}{2} \cdot (x - 2)

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

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Step 5.1.1.1

Rewrite.

y + 6 = 0 + 0 - \frac{3}{2} \cdot (x - 2)

$y + 6 = 0 + 0 - \frac{3}{2} \cdot (x - 2)$

Step 5.1.1.2

Simplify terms.

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Step 5.1.1.2.1

Apply the distributive property.

y + 6 = - \frac{3}{2} x - \frac{3}{2} \cdot - 2

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

Step 5.1.1.2.2

Combine

x

$x$ and

\frac{3}{2}

$\frac{3}{2}$ .

y + 6 = - \frac{x \cdot 3}{2} - \frac{3}{2} \cdot - 2

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

Step 5.1.1.2.3

Cancel the common factor of

2

$2$ .

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Step 5.1.1.2.3.1

Move the leading negative in

- \frac{3}{2}

$- \frac{3}{2}$ into the numerator.

y + 6 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot - 2

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

Step 5.1.1.2.3.2

Factor

2

$2$ out of

- 2

$- 2$ .

y + 6 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot (2 (- 1))

$y + 6 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot (2 (- 1))$

Step 5.1.1.2.3.3

Cancel the common factor.

$y + 6 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot (2 \cdot - 1)$

Step 5.1.1.2.3.4

Rewrite the expression.

$y + 6 = - \frac{x \cdot 3}{2} - 3 \cdot - 1$

Step 5.1.1.2.4

Multiply

$- 3$ by

$- 1$ .

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

Step 5.1.1.3

Move

$3$ to the left of

$x$ .

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

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

$6$ from both sides of the equation.

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

Step 5.1.2.2

Subtract

$6$ from

$3$ .

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

Step 5.2

Reorder terms.

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

Step 5.3

Remove parentheses.

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

Step 6