Question

the following system 4x - 5y = 20 (x - 2) + (y - 4) = 25 Betsy argues that the solution to the system above is only one ordered pair. Trent argues that there are two solutions, because when he graphed it, the graphs intersected in two points instead of one, as suggested by Betsy. Dora argues that the system has no solution because the graphs do not intersect at all. Determine which argument is correct and prove it graphically and algebraically, ​

Answer 1

The lines intersect at one point: (19.44, 11.56)

Explanation:

Rewrite both equations in standard format.

4x - 5y = 20

-5y = -4x + 20

y = (4/5)x - 4

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(x - 2) + (y - 4) = 25

y - 4 = 25 - (x-2)

y = -x + 2 + 25 + 4

y = -x + 31

We see that one slope is positive and the other is negative. Since these are straight line equations, the lines will cross at one point.

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We can find that point:

y = (4/5)x - 4

-x + 31 = (4/5)x - 4 [use y = -x + 31 from above]

-x - (4/5)x = -35

-(9/5)x = -35

x = (5/9)*(35)

x = 19.44

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y = -x + 31

y = -19.44 + 31

y = 11.56

(19.44, 11.56)

The graphical solution is attached.