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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.

Guest Dec 7, 2017
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\$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

CPhill  Dec 7, 2017

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