Finite Math Examples

$- 7 x - 5 y = 7$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $- 7 x - 5 y = 7$ .

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Step 2.1

Add $7 x$ to both sides of the equation.

$- 5 y = 7 + 7 x$

Step 2.2

Divide each term in $- 5 y = 7 + 7 x$ by $- 5$ and simplify.

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Step 2.2.1

Divide each term in $- 5 y = 7 + 7 x$ by $- 5$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 5$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $y$ by $1$ .

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

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

Move the negative in front of the fraction.

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

Step 2.2.3.1.2

Move the negative in front of the fraction.

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

Step 3

Find the slope when $y = - \frac{7}{5} - \frac{7 x}{5}$ .

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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 $- \frac{7}{5}$ and $- \frac{7 x}{5}$ .

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

Step 3.1.3

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

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Step 3.1.3.1

Reorder terms.

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

Step 3.1.3.2

Remove parentheses.

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

Step 3.2

Using the slope-intercept form, the slope is $- \frac{7}{5}$ .

$m = - \frac{7}{5}$

Step 4

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

$m_{perpendicular} = - \frac{1}{- \frac{7}{5}}$

Step 5

Simplify $- \frac{1}{- \frac{7}{5}}$ to find the slope of the perpendicular line.

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Step 5.1

Cancel the common factor of $1$ and $- 1$ .

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Step 5.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{7}{5}}$

Step 5.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{7}{5}}$

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{5}{7})$

Step 5.3

Multiply $\frac{5}{7}$ by $1$ .

$m_{perpendicular} = \frac{5}{7}$

Step 5.4

Multiply $- - \frac{5}{7}$ .

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Step 5.4.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{5}{7})$

Step 5.4.2

Multiply $\frac{5}{7}$ by $1$ .

$m_{perpendicular} = \frac{5}{7}$

Step 6

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

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Step 6.1

Use the slope $\frac{5}{7}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 6.2

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

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

Step 7

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

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Step 7.1

Solve for $y$ .

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Step 7.1.1

Add $y$ and $0$ .

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

Step 7.1.2

Simplify $\frac{5}{7} \cdot (x + 0)$ .

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Step 7.1.2.1

Add $x$ and $0$ .

$y = \frac{5}{7} \cdot x$

Step 7.1.2.2

Combine $\frac{5}{7}$ and $x$ .

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

Step 7.2

Reorder terms.

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

Step 8