Pre-Algebra Examples

$(7, 8)$ , $5 x + y = 7$

Step 1

Subtract $5 x$ from both sides of the equation.

$y = 7 - 5 x$

Step 2

Find the slope when $y = 7 - 5 x$ .

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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 $7$ and $- 5 x$ .

$y = - 5 x + 7$

Step 2.2

Using the slope-intercept form, the slope is $- 5$ .

$m = - 5$

Step 3

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

$m_{perpendicular} = - \frac{1}{- 5}$

Step 4

Simplify $- \frac{1}{- 5}$ to find the slope of the perpendicular line.

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Step 4.1

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{5}$

Step 4.2

Multiply $- - \frac{1}{5}$ .

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{1}{5})$

Step 4.2.2

Multiply $\frac{1}{5}$ by $1$ .

$m_{perpendicular} = \frac{1}{5}$

Step 5

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

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Step 5.1

Use the slope $\frac{1}{5}$ and a given point $(7, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = \frac{1}{5} \cdot (x - (7))$

Step 5.2

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

$y - 8 = \frac{1}{5} \cdot (x - 7)$

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 $\frac{1}{5} \cdot (x - 7)$ .

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Step 6.1.1.1

Rewrite.

$y - 8 = 0 + 0 + \frac{1}{5} \cdot (x - 7)$

Step 6.1.1.2

Simplify by adding zeros.

$y - 8 = \frac{1}{5} \cdot (x - 7)$

Step 6.1.1.3

Apply the distributive property.

$y - 8 = \frac{1}{5} x + \frac{1}{5} \cdot - 7$

Step 6.1.1.4

Combine $\frac{1}{5}$ and $x$ .

$y - 8 = \frac{x}{5} + \frac{1}{5} \cdot - 7$

Step 6.1.1.5

Combine $\frac{1}{5}$ and $- 7$ .

$y - 8 = \frac{x}{5} + \frac{- 7}{5}$

Step 6.1.1.6

Move the negative in front of the fraction.

$y - 8 = \frac{x}{5} - \frac{7}{5}$

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 $8$ to both sides of the equation.

$y = \frac{x}{5} - \frac{7}{5} + 8$

Step 6.1.2.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{x}{5} - \frac{7}{5} + 8 \cdot \frac{5}{5}$

Step 6.1.2.3

Combine $8$ and $\frac{5}{5}$ .

$y = \frac{x}{5} - \frac{7}{5} + \frac{8 \cdot 5}{5}$

Step 6.1.2.4

Combine the numerators over the common denominator.

$y = \frac{x}{5} + \frac{- 7 + 8 \cdot 5}{5}$

Step 6.1.2.5

Simplify the numerator.

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Step 6.1.2.5.1

Multiply $8$ by $5$ .

$y = \frac{x}{5} + \frac{- 7 + 40}{5}$

Step 6.1.2.5.2

Add $- 7$ and $40$ .

$y = \frac{x}{5} + \frac{33}{5}$

Step 6.2

Reorder terms.

$y = \frac{1}{5} x + \frac{33}{5}$

Step 7