Find Any Equation Perpendicular to the Line y/2+x=6

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

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $\frac{y}{2} + x = 6$ .

Subtract $x$ from both sides of the equation.

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

Multiply both sides of the equation by $2$ .

$2 \frac{y}{2} = 2 (6 - x)$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{y}{2} = 2 (6 - x)$

Rewrite the expression.

$y = 2 (6 - x)$

Simplify the right side.

Simplify $2 (6 - x)$ .

Apply the distributive property.

$y = 2 \cdot 6 + 2 (- x)$

Multiply.

Multiply $2$ by $6$ .

$y = 12 + 2 (- x)$

Multiply $- 1$ by $2$ .

$y = 12 - 2 x$

Reorder $12$ and $- 2 x$ .

$y = - 2 x + 12$

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

Simplify $- \frac{1}{- 2}$ to find the slope of the perpendicular line.

Move the negative in front of the fraction.

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

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

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{1}{2})$

Multiply $\frac{1}{2}$ by $1$ .

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

Step 6

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 7

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

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

Reorder terms.

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

Step 8

Find Any Equation Perpendicular to the Line y/2+x=6

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