Find the Perpendicular Line (5,-3) 4x-2y=7

$(5, - 3)$ $4 x - 2 y = 7$

Step 1

Solve $4 x - 2 y = 7$ .

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

$- 2 y = 7 - 4 x$

Divide each term in $- 2 y = 7 - 4 x$ by $- 2$ and simplify.

Divide each term in $- 2 y = 7 - 4 x$ by $- 2$ .

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Cancel the common factor of $- 4$ and $- 2$ .

Factor $- 2$ out of $- 4 x$ .

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

Cancel the common factors.

Factor $- 2$ out of $- 2$ .

$y = - \frac{7}{2} + \frac{- 2 (2 x)}{- 2 (1)}$

Cancel the common factor.

$y = - \frac{7}{2} + \frac{- 2 (2 x)}{- 2 \cdot 1}$

Rewrite the expression.

$y = - \frac{7}{2} + \frac{2 x}{1}$

Divide $2 x$ by $1$ .

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

Step 2

Find the slope when $y = - \frac{7}{2} + 2 x$ .

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 $- \frac{7}{2}$ and $2 x$ .

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

Using the slope-intercept form, the slope is $2$ .

$m = 2$

Step 3

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

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

Step 4

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

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

$y - (- 3) = - \frac{1}{2} \cdot (x - (5))$

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

$y + 3 = - \frac{1}{2} \cdot (x - 5)$

Step 5

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

Solve for $y$ .

Simplify $- \frac{1}{2} \cdot (x - 5)$ .

Rewrite.

$y + 3 = 0 + 0 - \frac{1}{2} \cdot (x - 5)$

Simplify by adding zeros.

$y + 3 = - \frac{1}{2} \cdot (x - 5)$

Apply the distributive property.

$y + 3 = - \frac{1}{2} x - \frac{1}{2} \cdot - 5$

Combine $x$ and $\frac{1}{2}$ .

$y + 3 = - \frac{x}{2} - \frac{1}{2} \cdot - 5$

Multiply $- \frac{1}{2} \cdot - 5$ .

Multiply $- 5$ by $- 1$ .

$y + 3 = - \frac{x}{2} + 5 (\frac{1}{2})$

Combine $5$ and $\frac{1}{2}$ .

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

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

Subtract $3$ from both sides of the equation.

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

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 3$ by $2$ .

$y = - \frac{x}{2} + \frac{5 - 6}{2}$

Subtract $6$ from $5$ .

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

Move the negative in front of the fraction.

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

Reorder terms.

$y = - (\frac{1}{2} x) - \frac{1}{2}$

Remove parentheses.

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

Step 6

Find the Perpendicular Line (5,-3) 4x-2y=7

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