Algebra Examples

$(3, - 2)$ that is perpendicular to the line $3 x + 4 y = 5$

Step 1

Solve $3 x + 4 y = 5$ .

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Step 1.1

Subtract $3 x$ from both sides of the equation.

$4 y = 5 - 3 x$

Step 1.2

Divide each term in $4 y = 5 - 3 x$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 y = 5 - 3 x$ by $4$ .

$\frac{4 y}{4} = \frac{5}{4} + \frac{- 3 x}{4}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{5}{4} + \frac{- 3 x}{4}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{5}{4} + \frac{- 3 x}{4}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$y = \frac{5}{4} - \frac{3 x}{4}$

Step 2

Find the slope when $y = \frac{5}{4} - \frac{3 x}{4}$ .

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Step 2.1

Rewrite in slope-intercept form.

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Step 2.1.1

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 2.1.2

Reorder $\frac{5}{4}$ and $- \frac{3 x}{4}$ .

$y = - \frac{3 x}{4} + \frac{5}{4}$

Step 2.1.3

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

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Step 2.1.3.1

Reorder terms.

$y = - (\frac{3}{4} x) + \frac{5}{4}$

Step 2.1.3.2

Remove parentheses.

$y = - \frac{3}{4} x + \frac{5}{4}$

Step 2.2

Using the slope-intercept form, the slope is $- \frac{3}{4}$ .

$m = - \frac{3}{4}$

Step 3

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

$m_{perpendicular} = - \frac{1}{- \frac{3}{4}}$

Step 4

Simplify $- \frac{1}{- \frac{3}{4}}$ to find the slope of the perpendicular line.

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Step 4.1

Cancel the common factor of $1$ and $- 1$ .

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Step 4.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{3}{4}}$

Step 4.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{3}{4}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{4}{3})$

Step 4.3

Multiply $\frac{4}{3}$ by $1$ .

$m_{perpendicular} = \frac{4}{3}$

Step 4.4

Multiply $- - \frac{4}{3}$ .

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Step 4.4.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{4}{3})$

Step 4.4.2

Multiply $\frac{4}{3}$ by $1$ .

$m_{perpendicular} = \frac{4}{3}$

Step 5

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

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Step 5.1

Use the slope $\frac{4}{3}$ and a given point $(3, - 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 5.2

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

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

Step 6

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

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Step 6.1

Solve for $y$ .

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Step 6.1.1

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

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Step 6.1.1.1

Rewrite.

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

Step 6.1.1.2

Simplify by adding zeros.

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

Step 6.1.1.3

Apply the distributive property.

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

Step 6.1.1.4

Combine $\frac{4}{3}$ and $x$ .

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

Step 6.1.1.5

Cancel the common factor of $3$ .

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Step 6.1.1.5.1

Factor $3$ out of $- 3$ .

$y + 2 = \frac{4 x}{3} + \frac{4}{3} \cdot (3 (- 1))$

Step 6.1.1.5.2

Cancel the common factor.

$y + 2 = \frac{4 x}{3} + \frac{4}{3} \cdot (3 \cdot - 1)$

Step 6.1.1.5.3

Rewrite the expression.

$y + 2 = \frac{4 x}{3} + 4 \cdot - 1$

Step 6.1.1.6

Multiply $4$ by $- 1$ .

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

Step 6.1.2

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

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Step 6.1.2.1

Subtract $2$ from both sides of the equation.

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

Step 6.1.2.2

Subtract $2$ from $- 4$ .

$y = \frac{4 x}{3} - 6$

Step 6.2

Reorder terms.

$y = \frac{4}{3} x - 6$

Step 7