Question 181781: Hi, I'm having trouble with the following questions:
For what value of 'a' is there no unique solution for the following?
3x - 2y = 5
5x + ay = 4
Also, when 'a' has this value, is there an infinite number of solutions or no solution? Justify your answer.
Thank you so much!
Answer by ptaylor(2198)   (Show Source):
You can put this solution on YOUR website!
3x - 2y = 5------eq1
5x + ay = 4 -----eq2
Put eq1 and eq2 in slope -intercept form (y=mx+b)
eq1:
-2y=5-3x divide each side by -2
y=(3/2)x-5/2-------------------------eq1 in slope-intercept form
eq2:
ay=4-5x divide each side by a
y=(-5/a)x+4/a--------------------------eq2 in slope-intercept form
If eq1 and eq2 have the same slope, they are either parallel (no solutions) or lay on top of each other (infinite solutions), so:
(-5/a)=3/2 multiply each side by 2a
-10=3a a=-10/3
Substitute a=-10/3 into eq2:
y=(-5/(-10/3)x+4/(-10/3) simplify
y=(3/2)x-6/5 ------------------------------eq2a
Now if we compare eq1 and eq2a, we see that they have the same slope but not the same y-intercept which means that if a=-(10/3) the lines are parallel and there is no solution.
Hope this helps---ptaylor
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