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# coordinate geometry

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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 ?

May 25, 2020

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Line t$$y= \frac{-3}{4}x+\frac{5}{4}$$

Perpendicular also means the "negative reciprocal of the slope" and in this case the negative reciprocal of the slope of Line t is $$\frac{4}{3}$$

Line p$$y=\frac{4}{3}x+?$$

To find the y-intercept, we need to put in the point (5, 5) into Line p.

$$5=\frac{4}{3}(5)+?$$ ---> $$5=\frac{20}{3}+?$$---> $$5-\frac{20}{3}=?$$---> $$\frac{-5}{3}=?$$

Line p$$y=\frac{4}{3}x+\frac{-5}{3}$$

To find the point common to both lines, we set them equal to each other:

$$\frac{4}{3}x+\frac{-5}{3}$$=$$\frac{-3}{4}x+\frac{5}{4}$$
$$x=\frac{7}{5}$$

Since we only need the "x" coordinate, $$x=\frac{7}{5}$$ is our answer.