Precalculus Examples

$7 x - 4 y = 5$ , $- x + 4 y = - 11$

Step 1

Find the slope and y-intercept of the first equation.

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Step 1.1

Rewrite in slope-intercept form.

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Step 1.1.1

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 1.1.2

Subtract $7 x$ from both sides of the equation.

$- 4 y = 5 - 7 x$

Step 1.1.3

Divide each term in $- 4 y = 5 - 7 x$ by $- 4$ and simplify.

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Step 1.1.3.1

Divide each term in $- 4 y = 5 - 7 x$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{5}{- 4} + \frac{- 7 x}{- 4}$

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $- 4$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{5}{- 4} + \frac{- 7 x}{- 4}$

Step 1.1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{5}{- 4} + \frac{- 7 x}{- 4}$

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Move the negative in front of the fraction.

$y = - \frac{5}{4} + \frac{- 7 x}{- 4}$

Step 1.1.3.3.1.2

Dividing two negative values results in a positive value.

$y = - \frac{5}{4} + \frac{7 x}{4}$

Step 1.1.4

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

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Step 1.1.4.1

Reorder $- \frac{5}{4}$ and $\frac{7 x}{4}$ .

$y = \frac{7 x}{4} - \frac{5}{4}$

Step 1.1.4.2

Reorder terms.

$y = \frac{7}{4} x - \frac{5}{4}$

Step 1.2

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m_{1} = \frac{7}{4}$

$b = - \frac{5}{4}$

$m_{1} = \frac{7}{4}$

$b = - \frac{5}{4}$

Step 2

Find the slope and y-intercept of the second equation.

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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

Add $x$ to both sides of the equation.

$4 y = - 11 + x$

Step 2.1.3

Divide each term in $4 y = - 11 + x$ by $4$ and simplify.

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Step 2.1.3.1

Divide each term in $4 y = - 11 + x$ by $4$ .

$\frac{4 y}{4} = \frac{- 11}{4} + \frac{x}{4}$

Step 2.1.3.2

Simplify the left side.

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Step 2.1.3.2.1

Cancel the common factor of $4$ .

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Step 2.1.3.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{- 11}{4} + \frac{x}{4}$

Step 2.1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 11}{4} + \frac{x}{4}$

Step 2.1.3.3

Simplify the right side.

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Step 2.1.3.3.1

Move the negative in front of the fraction.

$y = - \frac{11}{4} + \frac{x}{4}$

Step 2.1.4

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

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Step 2.1.4.1

Reorder $- \frac{11}{4}$ and $\frac{x}{4}$ .

$y = \frac{x}{4} - \frac{11}{4}$

Step 2.1.4.2

Reorder terms.

$y = \frac{1}{4} x - \frac{11}{4}$

Step 2.2

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m_{2} = \frac{1}{4}$

$b = - \frac{11}{4}$

$m_{2} = \frac{1}{4}$

$b = - \frac{11}{4}$

Step 3

Compare the slopes $m$ of the two equations.

$m_{1} = \frac{7}{4}, m_{2} = \frac{1}{4}$

Step 4

Compare the decimal form of one slope with the negative reciprocal of the other slope. If they are equal, then the lines are perpendicular. If the they are not equal, then the lines are not perpendicular.

$m_{1} = 1.75, m_{2} = - 4$

Step 5

The equations are not perpendicular because the slopes of the two lines are not negative reciprocals.

Not Perpendicular

Step 6