Algebra Examples

line $5 x - 4 y = 6$ , point $(- 4, 3)$

Step 1

Solve $5 x - 4 y = 6$ .

Tap for more steps...

Step 1.1

Subtract $5 x$ from both sides of the equation.

$- 4 y = 6 - 5 x$

Step 1.2

Divide each term in $- 4 y = 6 - 5 x$ by $- 4$ and simplify.

Tap for more steps...

Step 1.2.1

Divide each term in $- 4 y = 6 - 5 x$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{6}{- 4} + \frac{- 5 x}{- 4}$

Step 1.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{6}{- 4} + \frac{- 5 x}{- 4}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{6}{- 4} + \frac{- 5 x}{- 4}$

Step 1.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.2.3.1.1

Cancel the common factor of $6$ and $- 4$ .

Tap for more steps...

Step 1.2.3.1.1.1

Factor $2$ out of $6$ .

$y = \frac{2 (3)}{- 4} + \frac{- 5 x}{- 4}$

Step 1.2.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.1.1.2.1

Factor $2$ out of $- 4$ .

$y = \frac{2 \cdot 3}{2 \cdot - 2} + \frac{- 5 x}{- 4}$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot 3}{2 \cdot - 2} + \frac{- 5 x}{- 4}$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{3}{- 2} + \frac{- 5 x}{- 4}$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$y = - \frac{3}{2} + \frac{- 5 x}{- 4}$

Step 1.2.3.1.3

Dividing two negative values results in a positive value.

$y = - \frac{3}{2} + \frac{5 x}{4}$

Step 2

Find the slope when $y = - \frac{3}{2} + \frac{5 x}{4}$ .

Tap for more steps...

Step 2.1

Rewrite in slope-intercept form.

Tap for more steps...

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 $- \frac{3}{2}$ and $\frac{5 x}{4}$ .

$y = \frac{5 x}{4} - \frac{3}{2}$

Step 2.1.3

Reorder terms.

$y = \frac{5}{4} x - \frac{3}{2}$

Step 2.2

Using the slope-intercept form, the slope is $\frac{5}{4}$ .

$m = \frac{5}{4}$

Step 3

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

$m_{perpendicular} = - \frac{1}{\frac{5}{4}}$

Step 4

Simplify $- \frac{1}{\frac{5}{4}}$ to find the slope of the perpendicular line.

Tap for more steps...

Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{4}{5}))$

Step 4.2

Multiply $\frac{4}{5}$ by $1$ .

$m_{perpendicular} = - \frac{4}{5}$

Step 5

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

Tap for more steps...

Step 5.1

Use the slope $- \frac{4}{5}$ and a given point $(- 4, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 5.2

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

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

Step 6

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

Tap for more steps...

Step 6.1

Solve for $y$ .

Tap for more steps...

Step 6.1.1

Simplify $- \frac{4}{5} \cdot (x + 4)$ .

Tap for more steps...

Step 6.1.1.1

Rewrite.

$y - 3 = 0 + 0 - \frac{4}{5} \cdot (x + 4)$

Step 6.1.1.2

Simplify by adding zeros.

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

Step 6.1.1.3

Apply the distributive property.

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

Step 6.1.1.4

Combine $x$ and $\frac{4}{5}$ .

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

Step 6.1.1.5

Multiply $- \frac{4}{5} \cdot 4$ .

Tap for more steps...

Step 6.1.1.5.1

Multiply $4$ by $- 1$ .

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

Step 6.1.1.5.2

Combine $- 4$ and $\frac{4}{5}$ .

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

Step 6.1.1.5.3

Multiply $- 4$ by $4$ .

$y - 3 = - \frac{x \cdot 4}{5} + \frac{- 16}{5}$

Step 6.1.1.6

Simplify each term.

Tap for more steps...

Step 6.1.1.6.1

Move $4$ to the left of $x$ .

$y - 3 = - \frac{4 \cdot x}{5} + \frac{- 16}{5}$

Step 6.1.1.6.2

Move the negative in front of the fraction.

$y - 3 = - \frac{4 x}{5} - \frac{16}{5}$

Step 6.1.2

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

Tap for more steps...

Step 6.1.2.1

Add $3$ to both sides of the equation.

$y = - \frac{4 x}{5} - \frac{16}{5} + 3$

Step 6.1.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = - \frac{4 x}{5} - \frac{16}{5} + 3 \cdot \frac{5}{5}$

Step 6.1.2.3

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

$y = - \frac{4 x}{5} - \frac{16}{5} + \frac{3 \cdot 5}{5}$

Step 6.1.2.4

Combine the numerators over the common denominator.

$y = - \frac{4 x}{5} + \frac{- 16 + 3 \cdot 5}{5}$

Step 6.1.2.5

Simplify the numerator.

Tap for more steps...

Step 6.1.2.5.1

Multiply $3$ by $5$ .

$y = - \frac{4 x}{5} + \frac{- 16 + 15}{5}$

Step 6.1.2.5.2

Add $- 16$ and $15$ .

$y = - \frac{4 x}{5} + \frac{- 1}{5}$

Step 6.1.2.6

Move the negative in front of the fraction.

$y = - \frac{4 x}{5} - \frac{1}{5}$

Step 6.2

Reorder terms.

$y = - (\frac{4}{5} x) - \frac{1}{5}$

Step 6.3

Remove parentheses.

$y = - \frac{4}{5} x - \frac{1}{5}$

Step 7