Precalculus Examples

$4 x - 2 y = 7$ , $(2, 5)$

Step 1

Solve $4 x - 2 y = 7$ .

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Step 1.1

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

$- 2 y = 7 - 4 x$

Step 1.2

Divide each term in $- 2 y = 7 - 4 x$ by $- 2$ and simplify.

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Step 1.2.1

Divide each term in $- 2 y = 7 - 4 x$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{7}{- 2} + \frac{- 4 x}{- 2}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{7}{- 2} + \frac{- 4 x}{- 2}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{7}{- 2} + \frac{- 4 x}{- 2}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{7}{2} + \frac{- 4 x}{- 2}$

Step 1.2.3.1.2

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

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Step 1.2.3.1.2.1

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

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

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

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

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

Step 1.2.3.1.2.2.2

Cancel the common factor.

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

Step 1.2.3.1.2.2.3

Rewrite the expression.

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

Step 1.2.3.1.2.2.4

Divide $2 x$ by $1$ .

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

Step 2

Find the slope when $y = - \frac{7}{2} + 2 x$ .

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Step 2.1

Rewrite in slope-intercept form.

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Step 2.1.1

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 2.1.2

Reorder $- \frac{7}{2}$ and $2 x$ .

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

Step 2.2

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

$m = 2$

Step 3

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 4

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

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Step 4.1

Use the slope $- \frac{1}{2}$ and a given point $(2, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = - \frac{1}{2} \cdot (x - (2))$

Step 4.2

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

$y - 5 = - \frac{1}{2} \cdot (x - 2)$

Step 5

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

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Step 5.1

Solve for $y$ .

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Step 5.1.1

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

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Step 5.1.1.1

Rewrite.

$y - 5 = 0 + 0 - \frac{1}{2} \cdot (x - 2)$

Step 5.1.1.2

Simplify by adding zeros.

$y - 5 = - \frac{1}{2} \cdot (x - 2)$

Step 5.1.1.3

Apply the distributive property.

$y - 5 = - \frac{1}{2} x - \frac{1}{2} \cdot - 2$

Step 5.1.1.4

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

$y - 5 = - \frac{x}{2} - \frac{1}{2} \cdot - 2$

Step 5.1.1.5

Cancel the common factor of $2$ .

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Step 5.1.1.5.1

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

$y - 5 = - \frac{x}{2} + \frac{- 1}{2} \cdot - 2$

Step 5.1.1.5.2

Factor $2$ out of $- 2$ .

$y - 5 = - \frac{x}{2} + \frac{- 1}{2} \cdot (2 (- 1))$

Step 5.1.1.5.3

Cancel the common factor.

$y - 5 = - \frac{x}{2} + \frac{- 1}{2} \cdot (2 \cdot - 1)$

Step 5.1.1.5.4

Rewrite the expression.

$y - 5 = - \frac{x}{2} - 1 \cdot - 1$

Step 5.1.1.6

Multiply $- 1$ by $- 1$ .

$y - 5 = - \frac{x}{2} + 1$

Step 5.1.2

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

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Step 5.1.2.1

Add $5$ to both sides of the equation.

$y = - \frac{x}{2} + 1 + 5$

Step 5.1.2.2

Add $1$ and $5$ .

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

Step 5.2

Reorder terms.

$y = - (\frac{1}{2} x) + 6$

Step 5.3

Remove parentheses.

$y = - \frac{1}{2} x + 6$

Step 6