# Linear Algebra Solving a System of LE

• Workout
In summary, the student is trying to solve a system of equations but is not able to find a unique solution.
Workout
Okay here's my system of equations:

x − 3y − 2z = 0
−x + 2y + z = 0
2x + 4y + 6z = 0

Solve the following systems using Gaussian elimination

I put it in a matrix and did Gaussian elimination.

But I can't find a unique solution and it doesn't end up working out. Is this true? Or am I just making a mistake?

How can we tell without seeing what you did? Why would you expect a unique solution? Show us your work.

Workout said:
Okay here's my system of equations:

x − 3y − 2z = 0
−x + 2y + z = 0
2x + 4y + 6z = 0

Solve the following systems using Gaussian elimination

I put it in a matrix and did Gaussian elimination.

But I can't find a unique solution and it doesn't end up working out. Is this true? Or am I just making a mistake?
As LCKurtz said, show us what you did.

Also, don't blow away the homework template - it's there for a reason.

2R2 + R2 = | 1 -3 -2 0 |
-1 2 1 0
0 8 8 0

R1 + R2 = | 0 -3 -2 0 |
0 -1 -1 0
0 8 8 0

8R + R3 = | 1 -3 -2 0 |
0 -1 -1 0
0 0 0 0

So I was left with 2 equations.

x -3y -2z = 0
-y -z = 0

And I'm just stuck.

Basically when I solve it I get z = 0
then y = -z
and x = -z

There's no unique solution I guess. I was specifically looking for a unique solution. I think that was my problem.

So far, so good, but you can do just a bit more to completely reduce your matrix. You should end up here:

$$\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1\\ 0 & 0 & 0 \end{bmatrix}$$

This is shorthand for the system
x + z = 0
y + z = 0

Geometrically, what you have is two planes in space that intersect in a line.

Workout said:
Basically when I solve it I get z = 0
then y = -z
and x = -z

There's no unique solution I guess. I was specifically looking for a unique solution. I think that was my problem.
It really makes no sense to "get z= 0" and then have a solution that involves z!

It should have been obvious from the start that, since we have "= 0" on the right side of each equation (this was a "homogeneous" system), x= y= z= 0 is a solution. So if there had been a unique solution, it would have been trivial.

## 1. What is a system of linear equations (LE)?

A system of linear equations is a set of two or more equations that involve the same variables. The goal is to find the values of the variables that satisfy all of the given equations simultaneously.

## 2. How do you solve a system of linear equations?

There are several methods for solving a system of linear equations, including substitution, elimination, and graphing. These methods involve manipulating the equations to isolate one variable and then using that value to solve for the other variables.

## 3. Can a system of linear equations have more than one solution?

Yes, a system of linear equations can have one unique solution, infinitely many solutions, or no solution at all. The number of solutions depends on the number of variables and the number of equations in the system.

## 4. What is the importance of solving a system of linear equations?

Solving a system of linear equations is important in many fields, including mathematics, engineering, economics, and physics. It allows us to find the values of unknown variables and make predictions or solve real-world problems.

## 5. Is there a specific order in which equations should be solved in a system of linear equations?

No, there is no specific order in which equations should be solved. However, it is often more efficient to solve for the variable with the simplest coefficient or the variable that is easiest to isolate first.

Replies
2
Views
482
Replies
7
Views
1K
Replies
8
Views
1K
Replies
4
Views
614
Replies
2
Views
937
Replies
9
Views
1K
Replies
10
Views
1K
Replies
25
Views
3K
Replies
4
Views
2K
Replies
2
Views
580