+30 $A$ 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 $A+B?$
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
+2666 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 \(a + b = 6 + 52 = \color{brown}\boxed{58}\)
