Find Any Equation Perpendicular to the Line 4x+7y=8

$4 x + 7 y = 8$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $4 x + 7 y = 8$ .

Subtract $4 x$ from both sides of the equation.

$7 y = 8 - 4 x$

Divide each term in $7 y = 8 - 4 x$ by $7$ and simplify.

Divide each term in $7 y = 8 - 4 x$ by $7$ .

$\frac{7 y}{7} = \frac{8}{7} + \frac{- 4 x}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 y}{7} = \frac{8}{7} + \frac{- 4 x}{7}$

Divide $y$ by $1$ .

$y = \frac{8}{7} + \frac{- 4 x}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{8}{7} - \frac{4 x}{7}$

Step 3

Find the slope when $y = \frac{8}{7} - \frac{4 x}{7}$ .

Rewrite in slope-intercept form.

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Reorder $\frac{8}{7}$ and $- \frac{4 x}{7}$ .

$y = - \frac{4 x}{7} + \frac{8}{7}$

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

Reorder terms.

$y = - (\frac{4}{7} x) + \frac{8}{7}$

Remove parentheses.

$y = - \frac{4}{7} x + \frac{8}{7}$

Using the slope-intercept form, the slope is $- \frac{4}{7}$ .

$m = - \frac{4}{7}$

Step 4

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

$m_{perpendicular} = - \frac{1}{- \frac{4}{7}}$

Step 5

Simplify $- \frac{1}{- \frac{4}{7}}$ to find the slope of the perpendicular line.

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

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

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{4}{7}}$

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{4}{7}}$

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{7}{4})$

Multiply $\frac{7}{4}$ by $1$ .

$m_{perpendicular} = \frac{7}{4}$

Multiply $- - \frac{7}{4}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{7}{4})$

Multiply $\frac{7}{4}$ by $1$ .

$m_{perpendicular} = \frac{7}{4}$

Step 6

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

Use the slope $\frac{7}{4}$ and a given point $(0, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

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

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

Step 7

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

Solve for $y$ .

Add $y$ and $0$ .

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

Simplify $\frac{7}{4} \cdot (x + 0)$ .

Add $x$ and $0$ .

$y = \frac{7}{4} \cdot x$

Combine $\frac{7}{4}$ and $x$ .

$y = \frac{7 x}{4}$

Reorder terms.

$y = \frac{7}{4} x$

Step 8

Find Any Equation Perpendicular to the Line 4x+7y=8

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