Find Any Equation Perpendicular to the Line 4x-2y=1

$4 x - 2 y = 1$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $4 x - 2 y = 1$ .

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

$- 2 y = 1 - 4 x$

Divide each term in $- 2 y = 1 - 4 x$ by $- 2$ and simplify.

Divide each term in $- 2 y = 1 - 4 x$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{1}{- 2} + \frac{- 4 x}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{1}{- 2} + \frac{- 4 x}{- 2}$

Divide $y$ by $1$ .

$y = \frac{1}{- 2} + \frac{- 4 x}{- 2}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{1}{2} + \frac{- 4 x}{- 2}$

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2 x$ by $1$ .

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

Step 3

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

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

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

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 4x-2y=1

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