Find Any Equation Perpendicular to the Line 2y-x=1

$2 y - x = 1$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $2 y - x = 1$ .

Add $x$ to both sides of the equation.

$2 y = 1 + x$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Step 3

Find the slope when $y = \frac{1}{2} + \frac{x}{2}$ .

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

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

Reorder terms.

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

Using the slope-intercept form, the slope is $\frac{1}{2}$ .

$m = \frac{1}{2}$

Step 4

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

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

Step 5

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

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 \cdot 2)$

Multiply $- (1 \cdot 2)$ .

Multiply $2$ by $1$ .

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

Multiply $- 1$ by $2$ .

$m_{perpendicular} = - 2$

Step 6

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

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

$y - (0) = - 2 \cdot (x - (0))$

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

$y + 0 = - 2 \cdot (x + 0)$

Step 7

Solve for $y$ .

Add $y$ and $0$ .

$y = - 2 \cdot (x + 0)$

Add $x$ and $0$ .

$y = - 2 x$

Step 8

Find Any Equation Perpendicular to the Line 2y-x=1

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