Question

Is it possible for a linear system to have a unique solution if it has more equations than variables? If yes, give an example. If no, justify why it is impossible.

Answer 1

Yes, it's possible.

Explanation:

We know that a linear system has a unique solution if it has the same number of equations than variables (but remember that you musn't have a linear combination of equations).

So it is possible to create a system with more equations than variables, with a unique solution.

Example:

1) X + Y + Z = 3

2) 2X + 2Y + 2Z = 6

3) X + Y = 2

4) X + Z = 2

This is a 4 equation and 3 variable system. Pay attention to equation number 1) and 2). They are a linear combination (number 2 is the number 1 multiplied by 2). So, for this system you can basically ignore one of them.

If you do that, you will have 3 equations and 3 variables, with a unique solution. In this case solution is X = 1, Y = 1, Z = 1.

Now, let's see if equation number 2 satisfies:

2x1 + 2x1 + 2x1 = 6

6 = 6

With this simple example we proved that you can have a linear system with a unique solution if it has more equations than variables.

Answer 2

Yes

Explanation:

Yes it is possible, if the equations are equivalent. For example in one variable, we have x=2 and if we add the equation 2x=4 both considered on the Real numbers, then the system has two equations and one variable, more equations than variables and yet the solution is unique x=2.