How many solutions are there to the following systems of equations: 2x-y=2, -x+5y=3?

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Answer 1 · 2017-02-08T00:05:39

The only solution is #x=13/9, y=8/9#

I like to write both equations in slope-intercept form for easier comparison:

#2x-y=2# becomes #y=2x-2# slope = 2, #y#-intercept = -2

#-x+5y=3# becomes #y=1/5x+3/5# slope = #1/5#, #y#-intercept=#3/5#

The lines are not parallel. This would require the slopes to be equal.

(They also are not the same line, as this would require the same slope and intercept.)

Therefore, there must be a single, unique point of intersection of the two lines. This intersection point is the solution to the equations.

To find it, we can set the right sides of the two slope-intercept equations equal to each other (as they both equal #y#)

#2x-2=1/5x+3/5#

#2x-1/5x=3/5+2#

Multiply every term by five:

#10x-x=3+10#

#9x=13#

#x=13/9#

Since #y=2x-2#

#y=2(13/9)-2 = 26/9 - 18/9 = 8/9#

The solution is #x=13/9, y=8/9#