Algebra Examples

What is an equation of the line that passes through the point $(4, 1)$ and is perpendicular to the line $2 x - y = 4$ ?

Step 1

Write the problem as a mathematical expression.

$(4, 1)$ , $2 x - y = 4$

Step 2

Solve $2 x - y = 4$ .

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Step 2.1

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

$- y = 4 - 2 x$

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2.2

Divide $y$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Divide $4$ by $- 1$ .

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

Step 2.2.3.1.2

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

$y = - 4 - 1 \cdot (- 2 x)$

Step 2.2.3.1.3

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

$y = - 4 - (- 2 x)$

Step 2.2.3.1.4

Multiply $- 2$ by $- 1$ .

$y = - 4 + 2 x$

Step 3

Find the slope when $y = - 4 + 2 x$ .

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Step 3.1

Rewrite in slope-intercept form.

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Step 3.1.1

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

$y = m x + b$

Step 3.1.2

Reorder $- 4$ and $2 x$ .

$y = 2 x - 4$

Step 3.2

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.

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Step 5.1

Use the slope $- \frac{1}{2}$ and a given point $(4, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = - \frac{1}{2} \cdot (x - (4))$

Step 5.2

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

$y - 1 = - \frac{1}{2} \cdot (x - 4)$

Step 6

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

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Step 6.1

Solve for $y$ .

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Step 6.1.1

Simplify $- \frac{1}{2} \cdot (x - 4)$ .

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Step 6.1.1.1

Rewrite.

$y - 1 = 0 + 0 - \frac{1}{2} \cdot (x - 4)$

Step 6.1.1.2

Simplify by adding zeros.

$y - 1 = - \frac{1}{2} \cdot (x - 4)$

Step 6.1.1.3

Apply the distributive property.

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

Step 6.1.1.4

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

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

Step 6.1.1.5

Cancel the common factor of $2$ .

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Step 6.1.1.5.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 6.1.1.5.2

Factor $2$ out of $- 4$ .

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

Step 6.1.1.5.3

Cancel the common factor.

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

Step 6.1.1.5.4

Rewrite the expression.

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

Step 6.1.1.6

Multiply $- 1$ by $- 2$ .

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

Step 6.1.2

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

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Step 6.1.2.1

Add $1$ to both sides of the equation.

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

Step 6.1.2.2

Add $2$ and $1$ .

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

Step 6.2

Reorder terms.

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

Step 6.3

Remove parentheses.

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

Step 7