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Linear functions are a family of functions that all have a constant rate of change. There are additional ways that pairs of lines can relate. One such way is if the lines are perpendicular. In this section, what makes two lines perpendicular and the ways in which their rules and graphs are similar/different will be explored.

$m_{1}⋅m_{2}=-1$

This graph shows one pair of perpendicular lines.

For the product of two slopes to equal $-1,$ the slopes must be opposite reciprocals. For example, $m_{1}=-32 $ and $m_{2}=23 $ are opposite reciprocal slopes.

Are the lines shown in the graph perpendicular? Justify your answer.

Show Solution

Two lines are perpendicular if the product of their slopes is $-1.$ Notice that the equations are given in slope-intercept form. Thus, we can see that the lines have the slopes $m_{1}=-2andm_{2}=52 .$
Let's multiply the slopes to determine if their product is $-1.$
The product of the slopes does not equal $-1.$ Therefore, the lines are **not** perpendicular.

$m_{1}⋅m_{2}=-1$

SubstituteII

$m_{1}=-2$, $m_{2}=52 $

$-2⋅52 =?-1$

MoveLeftFacToNum

$a⋅cb =ca⋅b $

$5-2⋅2 =?-1$

Multiply

Multiply

$5-4 =-1$

Write the equation of the line that's perpendicular to $y=32 x−1$ and passes through the point $(0,-1).$

Show Solution

To write the equation of the line in slope-intercept form, we need its slope and its $y$-intercept. To ensure the lines are perpendicular, their slopes, $m_{1}$ and $m_{2},$ must multiply to equal $-1.$ $m_{1}⋅m_{2}=-1$
The given line has the slope $m_{1}=32 .$ Thus, we can use the product to determine $m_{2}.$
The slope of the other line is $m=-23 ,$ so we can write the incomplete equation of this line as $y=-23 x+b.$
Recall that the line must pass through the point $(0,-1).$ To find the $y$-intercept, we can substitute the point $(0,-1)$ for $x$ and $y$ in the incomplete equation above.
The $y$-intercept of the line is $b=-1.$ This gives the equation
$y=-23 x−1.$

$m_{1}⋅m_{2}=-1$

Substitute

$m_{1}=32 $

$32 ⋅m_{2}=-1$

MultEqn

$LHS⋅3=RHS⋅3$

$2⋅m_{2}=-3$

DivEqn

$LHS/2=RHS/2$

$m_{2}=-23 $

$y=-23 x+b$

SubstituteII

$x=0$, $y=-1$

$-1=-23 ⋅0+b$

ZeroPropMult

Zero Property of Multiplication

$-1=b$

RearrangeEqn

Rearrange equation

$b=-1$

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