SOLUTION: Use the following: 2/3x+y=16 and kx+3y=48 For what values of k does the linear system below have: a) infinite solutions? b) one solution? c) no solution?

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Question 1076487: Use the following: 2/3x+y=16 and kx+3y=48
For what values of k does the linear system below have:
a) infinite solutions?
b) one solution?
c) no solution?
Found 2 solutions by ikleyn, Boreal:
Answer by ikleyn(52876)   (Show Source): You can put this solution on YOUR website!
.
2/3x +  y = 16 
  kx + 3y = 48


For what values of k does the linear system below have:

a) infinitely many solutions?    - At k = 2.  Then the equations and right hand sides are proportional, and the system is DEPENDENT.

b) one solution?                 - At any other value of k =/= 2.

c) no solution?                  - There is NO such a value of k.


Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
(2/3)x+y=16
kx+3y=48
if kx=2, then 2x+3y=48 and multiplying the top by 3, 2x+3y=48, the same.
Therefore k=2 for infinite solutions.
The slope of the first is -2/3, and slope intercept form is -2x/3+16
the slope of the second is -(1/3)k and slope intercept form is -k/3+16
For k equal to any other number than 2, there will be one point of intersection.
For no solution, the two lines have to be parallel, and the line kx+3y=48 becomes 3y=-kx+48 and
y=-(k/3)+16. The y-intercept doesn't change with k, so there is no value where the lines are parallel without being the same line.
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