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$A$ and $B$ are constants such that the graphs of the lines $3x - 4y = 7$ and $8x + Ay = B$ are perpendicular and intersect at $(5,2).$ What is $A+B?$

 Jul 26, 2022
 #1
avatar
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3x - 4y= 7

8x + Ay = B

 

The slope of the first line is 3/4

So the slope of the second line will be  -4/3 

 

So we can write the second line as

 

Ay  =  -8x + B  ⇒  y  =  (-8/A)x + B/A

 

So this implies that   

 

(-8/A)  = -4/3

 

A /-8  =  -3/4

 

A = 24/ 4   = 6

 

So we have that

 

2 =  (-4/3) (5) + B/6

 

2 = -20/3 + B/6

 

12 = -40 + B

 

52  = B

 

So  A + B =    6 + 52  =  58

 Jul 26, 2022
 #3
avatar+1155 
+5

you just copied CPhill's answer!!!

nerdiest  Jul 26, 2022
 #2
avatar+2448 
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Find the slope of equation 1 as follows: 

 

\(3x - 4y = 7 \)

\(-4y = 7 - 3x \)

\(4y = -7 + 3x \)

\(4y = 3x - 7\)
\(y = {3 \over 4} x - {7 \over 4}\)

 

The slope of the line is \({3 \over 4}\), so the slope of the other line is \(-{4 \over 3}\)(negative reciprocal)

 

Now, convert the second equation into slope-intercept form:

 

 \(8x + Ay = B\)

\(Ay = B - 8x\)

\(y = {B \over A} - {8 \over A}x\)

\(y = -{8 \over A} x + {B \over A}\)

 

The slope of this line is \({-8 \over A}\), and we can solve for A: \(-{8 \over A} = -{4 \over 3} \Rightarrow a = 6\)

 

Now, plugging this back into the equation gives us: \(8x + 6y = B\)

 

Plug in the given coordinates: \(8 \times 5 + 6 \times 2 = 52\)

 

So, \(a = 6\) and \(b = 52\), meaning \(a + b = 6 + 52 = \color{brown}\boxed{58}\)

 Jul 26, 2022

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