Algebra Examples

$(4, 7)$ $y = \frac{2}{3} x - 1$

Step 1

Find the slope when $y = \frac{2}{3} x - 1$ .

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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 $x$ .

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

Step 1.1.3

Reorder terms.

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

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 $(4, 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (7) = - \frac{3}{2} \cdot (x - (4))$

Step 4.2

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

$y - 7 = - \frac{3}{2} \cdot (x - 4)$

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 $- \frac{3}{2} \cdot (x - 4)$ .

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Step 5.1.1.1

Rewrite.

$y - 7 = 0 + 0 - \frac{3}{2} \cdot (x - 4)$

Step 5.1.1.2

Simplify terms.

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Step 5.1.1.2.1

Apply the distributive property.

$y - 7 = - \frac{3}{2} x - \frac{3}{2} \cdot - 4$

Step 5.1.1.2.2

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

$y - 7 = - \frac{x \cdot 3}{2} - \frac{3}{2} \cdot - 4$

Step 5.1.1.2.3

Cancel the common factor of $2$ .

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Step 5.1.1.2.3.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$y - 7 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot - 4$

Step 5.1.1.2.3.2

Factor $2$ out of $- 4$ .

$y - 7 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot (2 (- 2))$

Step 5.1.1.2.3.3

Cancel the common factor.

$y - 7 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot (2 \cdot - 2)$

Step 5.1.1.2.3.4

Rewrite the expression.

$y - 7 = - \frac{x \cdot 3}{2} - 3 \cdot - 2$

Step 5.1.1.2.4

Multiply $- 3$ by $- 2$ .

$y - 7 = - \frac{x \cdot 3}{2} + 6$

Step 5.1.1.3

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

$y - 7 = - \frac{3 x}{2} + 6$

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 $7$ to both sides of the equation.

$y = - \frac{3 x}{2} + 6 + 7$

Step 5.1.2.2

Add $6$ and $7$ .

$y = - \frac{3 x}{2} + 13$

Step 5.2

Reorder terms.

$y = - (\frac{3}{2} x) + 13$

Step 5.3

Remove parentheses.

$y = - \frac{3}{2} x + 13$

Step 6