Math Help - system of equations problem

1. system of equations problem

Solve the following system of equations.

$
x_1-x_2+x_3=-5
$

$
3x_1+x_2-x_3=25
$

$
2x_1+x_2+3x_3=20
$

$
6x_1+x_2+3x_3=40
$

Does this system has a unique solution or infinitely many solutions? Justify your answer.

please explain to me how to do it, thank you!

2. This is a very standard problem. Do you have a textbook? It should have examples...
I'd be surprised if you were asked to solve this without first being given the proper tools.

3. Originally Posted by Bruno J.
This is a very standard problem. Do you have a textbook? It should have examples...
I'd be surprised if you were asked to solve this without first being given the proper tools.
It doesn't look so standard to me... four equations in three unknowns...

4. Originally Posted by Bruno J.
This is a very standard problem. Do you have a textbook? It should have examples...
I'd be surprised if you were asked to solve this without first being given the proper tools.
we've already studied for 3 equations, not 4

5. Originally Posted by quah13579
Solve the following system of equations.

$
x_1-x_2+x_3=-5
$

$
3x_1+x_2-x_3=25
$

$
2x_1+x_2+3x_3=20
$

$
6x_1+x_2+3x_3=40
$

Does this system has a unique solution or infinitely many solutions? Justify your answer.

please explain to me how to do it, thank you!
Solve for $x_1$ by adding equations (1) and (2).

Solve for $x_1$ by subtracting equation (3) from equation (4).

What do you conclude?

6. Originally Posted by mr fantastic
Solve for $x_1$ by adding equations (1) and (2).

Solve for $x_1$ by subtracting equation (3) from equation (4).

What do you conclude?
I got $x_1$ = 5. so it is a unique solution right?

7. it should be unique
since there are more equations then unknowns, correct?

8. Originally Posted by mpl06c
it should be unique
since there are more equations then unknowns, correct?
Incorrect.

In the case of the question posted, there is a unique solution. However, when there are more equations then unknowns it is still possible for equations to be:

1. Inconsistent (in which case there is no solution). eg. x + y = 3, 2x + 2y = -1, x - y = 2 (three equations, two unknowns, no solution)

2. Redundant (in which case there may be infinite solutions). eg. x + y = 3, 2x + 2y = 6, -x - y = -3 (three equations, two unknowns, infinite solutions).

9. oh ok... so in order to tell if it was uniqiue one would have to do RREF, correct?