SOLUTION: Hello, I have linear equation problem that I am stuck on. I have tried the problem myself but I'm having trouble with it. Here is the problem I'm working on: Determine the nece

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Question 398712: Hello,
I have linear equation problem that I am stuck on. I have tried the problem myself but I'm having trouble with it. Here is the problem I'm working on:
Determine the necessary conditions on a, b and c for the following systems to have:
a unique solution; an infinite number of solutions; or be inconsistent.
x1 + ax2 = 5
3x1 + 6x2 = b
Here is the work I've done so far, and this is all in matrices so imagine the boxes around the numbers:
1 a |5
3 6 |b
multiplying row 2 by 1/3
1 a | 5
1 2 | b/3
subtracting row 1 from row 2
1 a |5
0 2-a |b/3 - 5
I'm not sure if what I've done is correct, but this is where I'm stuck because I do not know how to set up the three conditions?
Thank you for any help
Yury
Answer by stanbon(57347) (Show Source):
You can put this solution on YOUR website!
Determine the necessary conditions on a, b and c for the following systems to have:
a unique solution; an in finite number of solutions; or be inconsistent.
x1 + ax2 = 5
3x1 + 6x2 = b
---
Put in slope-intercept form:
x1 = -ax2 + 5
slope = -a
int = 5
---------
x1 = -2x2 - (b/3)
slope = -2
int = (-b/3)
============================
Unique solution when slopes are not equal:
-a is not equal to -2
a is not equal to 2
--------------------
Inconsistent when slopes are equal.
-a = -2
a = 2
---------------
Infinite # of solutions when slopes and int are equal.
a=2
and (-b/3) = 5
b = -15
===================
Cheers,
Stan H.