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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 $(5,2).$ Wha

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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

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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
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nerdiest  Jul 26, 2022
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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