Determine if Perpendicular Are the lines defined by the equations y=3x+1 and y=-1/3x-5 perpendicular?

Are the lines defined by the equations $y = 3 x + 1$ and $y = - \frac{1}{3} x - 5$ perpendicular?

Step 1

Use the slope-intercept form to find the slope and y-intercept.

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

$y = m x + b$

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

$m_{1} = 3$

$b = 1$

$m_{1} = 3$

$b = 1$

Step 2

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

Rewrite in slope-intercept form.

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

$y = m x + b$

Simplify the right side.

Combine $x$ and $\frac{1}{3}$ .

$y = - \frac{x}{3} - 5$

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

Reorder terms.

$y = - (\frac{1}{3} x) - 5$

Remove parentheses.

$y = - \frac{1}{3} x - 5$

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

$m_{2} = - \frac{1}{3}$

$b = - 5$

$m_{2} = - \frac{1}{3}$

$b = - 5$

Step 3

Compare the slopes $m$ of the two equations.

$m_{1} = 3, m_{2} = - \frac{1}{3}$

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.

$m_{1} = 3, m_{2} = 3$

Step 5

The equations are perpendicular because the slopes of the two lines are negative reciprocals.

Perpendicular

Step 6

Determine if Perpendicular Are the lines defined by the equations y=3x+1 and y=-1/3x-5 perpendicular?

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