Finding The Constants a,b and c.

[Bashyboy]

Homework Statement

In Exercises 49 and 50, find values of a, b, and c(if possible) such that the system of linear equations has (a) a unique solution, (b) no solution, and (c) an infinite number of solutions.

49.

x+y ~ = 2

~y+z =2

x+ ~ z =2

ax + by + cz=0

Homework Equations

The Attempt at a Solution

I immediately recognized this system as one consisting of planes. Hence, wouldn't it be impossible for there to exist a unique solution, as my geometric intuition of the situation ?

 

[hilbert2]

I immediately recognized this system as one consisting of planes. Hence, wouldn't it be impossible for there to exist a unique solution, as my geometric intuition of the situation ?

What about case a=1, b=-1, c=0 ? Isn't there a unique solution?

 

[HallsofIvy]

No, it is quite possible for four planes to intersect in a single point. What makes you think otherwise?

(Boy, I have to learn to type faster!)

 

[Bashyboy]

Well, I was considering the case of two planes. Would you agree that it is impossible for there to exist a unique solution, when considering two planes?

 

[HallsofIvy]

No, that's not enough. You need at least three planes. And 4 is larger than 3!

What you can do is solve the first three equations for the unique point of intersection, then see what a, b, and c must be in order that those values of x, y, and z satisfy the final equation.

 

[Bashyboy]

HallsofIvy, I am confused, what do you mean by, "No, that's not enough. You need at least three planes. And 4 is larger than 3!" what are you alluding to?

 

[haruspex]

Well, I was considering the case of two planes. Would you agree that it is impossible for there to exist a unique solution, when considering two planes?

That's right - with just two planes you must have no solutions, or a line of solutions, or a plane of solutions.
Considering just the first three planes you're given, how many solutions are there?

 

[Bashyboy]

Haruspex, I took the first three equations and came up with this:

(1) x+y=2

(2) y+z=2

(3) x+z=2

Rearranging (1), y=2-x, and substituting the result into (2): (2-x) +z=2, which becomes z-x=0; adding this equation to (3): 2z=0 \implies z=0

If z=0, then y=2 and x = 0.

If the fourth equation was equal to two, rather than 0, then I could choose the constants such that it would be identical to one of the other planes.

 

[Bashyboy]

Hold on, I think I figured it out. Let me type my response.

 

[Bashyboy]

For the first three exists, there exists a unique solution, namely, (0,2,0). We need to retain this uniqueness, when dealing with the fourth equation, and to do so, (0,2,0) must satisfy the fourth equation, and we must choose constants that are conducive to this. a(0) + b(-2) + c(0) = 0. From this, it is clear that, a and c can be any real numbers, but b must be zero. Does this sound correct?

 

[D H]

For the first three exists, there exists a unique solution, namely, (0,2,0).

Does this satisfy x+z=2? You might want to try again.

 

[haruspex]

z-x=0; adding this equation to (3): 2z=0 \implies z=0

Retry those steps.

 

[Bashyboy]

Oh, wait. There should have been a 2 on the right-hand side of the equation, meaning that z=1, y=1, and x=1.

 

[D H]

Much better.

 

[Bashyboy]

For part (b), to make the system inconsistent, must I chose constants that will guarantee the equation a + b + c = 0 is false? I'm not sure how I will solve part (c), yet.

 

[HallsofIvy]

Yes, start from the first three equations: Subtracting y+ z= 2 from x+ y= 2, the "y" terms cancel and we have x- z= 0. Adding that to x+ z= 2, the "z" terms cancel and we have 2x= 2 so that x= 1. Then x+ z= 1+ z= 2 so that z= 1 and x+y= 1+ y= 2 so y= 1. The solution to the first three equations is x= y= z= 1.

NOW look at "ax+ by+ cz= 1". With x= y= z= 1, that gives a+ b+ c= 1. As long as that is true, the four equations will have the unique solution x= y= z= 1. If it is not true, there will be no solution. Since x= y= z= 1 is the only solution to the first three equations, there is no way to have more than one solution.