Let line $t$ be the line represented by $3x+4y = 5$ and let line $p$ be the line perpendicular to line $t$ and containing the point $(5,5)$. What is the $x$-coordinate of the point common to line $t$ and line $p$? Express your answer as a common fraction.
$3x+4y = 5 ⇒ y = (-3/4)x + 5/4 (1)
The slope of this line = -3/4
So....the slope of a perpendicular line = 4/3
And the equation of thiis line is
y = (4/3)(x - 5) + 5
y = (4/3)x - 20/3 + 5
y = (4/3)x -5/3 (2)
Setting (1) and (2) equal, we have that
(-3/4)x + 5/4 = (4/3)x - 5/3 multiply through by 12
-9x + 15 = 16x - 20 add 20, 9x to both sides
35 = 25x divide both sides by 25
35/25 = 7/5 = x