SOLUTION: Hi, I'm Michele and I could really use someones help on my problem. I will be greatful and very thankful for who ever will help me. Thank You for your help. 1.Find k if the foll

Question 144078: Hi, I'm Michele and I could really use someones help on my problem. I will be greatful and very thankful for who ever will help me. Thank You for your help.
1.Find k if the following system of equations has a unique solution
2x + (k - 1)y = 6
3x + (2k + 1)y = 9

2.Find k if the following system of equations has infinite solutions
kx + 3y = k - 3
12x + ky = k

3.Find k if the following system of equations has no solution
3x + 2y = 6
kx + (k - 1)y = 9

Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
1.Find k if the following system of equations has a unique solution
2x + (k - 1)y = 6
3x + (2k + 1)y = 9
------------------
Multiply thru 1st by 3 ; Multiply thru 2nd by 2 to get:
6x + (3k-3)y = 18
6x + (4k+2)y = 18
--------------------
Subtract 1st from 2nd to get:
[4k+2-(3k-3)]y = 0
[k+5] y = 0
Comment: y is a variable so may not always be zero
k+5 = 0 when k=-5
As long as k is not -5, y will be zero and x will be 3
So the system will have a unique solution as long as k is not equal to -5.
==================================
2.Find k if the following system of equations has infinite solutions
kx + 3y = k - 3
12x + ky = k
-------
The system has infinite solutions if the slopes and the intercepts are the same
Put them in slope-intercept form:
y = (-k/3)x + (k-3)/3
y = (-12/k)x + 1
----------
Equating slopes: -k/3 = -12/k ; k = 6
Equating intercepts: (k-3)/3 = 1 ; k = 6
Conclusion: infinitely many solutions if k = 6
-----------------------------------------
Find k if the following system of equations has no solution
3x + 2y = 6
kx + (k - 1)y = 9
--------
No solutions if slopes are the same and intercepts are not.
y = (-3/2)x + 3
y = (-k/(k-1)x + 9/(k-1)
------------------------
Equating slopes: -3/2 = -k/(k-1) ; -2k = -3k+3 ; k = 3
Equatiing intercepts: 3 = 9/(k-1) ; k-1 = 3 ; k = 4
-------------
Conclusion: No solutions if k = 3
=====================================
Cheers,
Stan H.

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