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Find the acute angle between the lines. Round your answer to the nearest degree. 5x − y = 2, 2x + y = 7

 Dec 9, 2021
 #1
avatar+203 
+1

Let's find the slope of each line

 

y = 9x - 3         y = -8 + 8

 

 

 

And the slope of the first line is 9

 

And the slope of the second line is -8

 

 

 

And taking the tangent inverse of the second line, we have

 

 

 

tan-1(-8)  = -82.87°   We need to add 180° to this to get the correct angle = 97.13°

 

 

 

And taking the tangent inverse of the first line, we have

 

 

 

tan-1(9) = 83.66°

 

 

 

So.....subtrating the second result from the first, we have 13.47° = about 13°

 Dec 9, 2021
 #2
avatar+132 
+1

Hi AlgebraGuru,

 

I believe you have made a mistake, the question says 5x − y = 2, 2x + y = 7, not y = 9x - 3, y = -8x + 8.

This can be solved with trig and coordinate bashing (I got the impression that vectors may be the easier way). I graphed the red and blue lines as the lines in the problem, and the green line is a random line that's perpendicular to blue. The red and blue lines intersect at (9/7, 31/7); the green line intersects the red line at (4/9,2/9) and the blue at (14/5, 7/5). I used wolframalpha for the steps, but essentially you find all of the sides using distance formula, apply any one of the trig functions, then solve for the angle using inverse trig function. The result when we do this is 37.875 ≈ 38˚

 

Update: I just discovered that the formula is arctan((slope_1 + slope_2)/(1+slope_1*slope_2)) in degrees which gives the same answer :|

(now I wanna prove it 😅)

tinfoilhat  Dec 9, 2021

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