Find Any Equation Perpendicular to the Line 2x-y-7=0

$2 x - y - 7 = 0$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $2 x - y - 7 = 0$ .

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

Subtract $2 x$ from both sides of the equation.

$- y - 7 = - 2 x$

Add $7$ to both sides of the equation.

$- y = - 2 x + 7$

Divide each term in $- y = - 2 x + 7$ by $- 1$ and simplify.

Divide each term in $- y = - 2 x + 7$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{- 2 x}{- 1}$ .

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

Rewrite $- 1 \cdot (- 2 x)$ as $- (- 2 x)$ .

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

Multiply $- 2$ by $- 1$ .

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

Divide $7$ by $- 1$ .

$y = 2 x - 7$

Step 3

Use the slope-intercept form to find the slope.

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

$y = m x + b$

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

$m = 2$

Step 4

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

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

Step 5

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

Use the slope $- \frac{1}{2}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

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

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

Step 6

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

Solve for $y$ .

Add $y$ and $0$ .

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

Simplify $- \frac{1}{2} \cdot (x + 0)$ .

Add $x$ and $0$ .

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

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

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

Reorder terms.

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

Remove parentheses.

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

Step 7

Find Any Equation Perpendicular to the Line 2x-y-7=0

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