# system of equations problem

• Sep 20th 2009, 08:25 PM
quah13579
system of equations problem
Solve the following system of equations.

\$\displaystyle
x_1-x_2+x_3=-5
\$
\$\displaystyle
3x_1+x_2-x_3=25
\$
\$\displaystyle
2x_1+x_2+3x_3=20
\$
\$\displaystyle
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!
• Sep 20th 2009, 09:26 PM
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.
• Sep 20th 2009, 10:03 PM
Prove It
Quote:

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...
• Sep 20th 2009, 10:30 PM
quah13579
Quote:

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
• Sep 20th 2009, 11:51 PM
mr fantastic
Quote:

Originally Posted by quah13579
Solve the following system of equations.

\$\displaystyle
x_1-x_2+x_3=-5
\$
\$\displaystyle
3x_1+x_2-x_3=25
\$
\$\displaystyle
2x_1+x_2+3x_3=20
\$
\$\displaystyle
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 \$\displaystyle x_1\$ by adding equations (1) and (2).

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

What do you conclude?
• Sep 21st 2009, 12:26 AM
quah13579
Quote:

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

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

What do you conclude?

I got \$\displaystyle x_1\$ = 5. so it is a unique solution right?
• Sep 21st 2009, 06:36 PM
mpl06c
it should be unique
since there are more equations then unknowns, correct?
• Sep 21st 2009, 06:46 PM
mr fantastic
Quote:

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).
• Sep 21st 2009, 07:05 PM
mpl06c
oh ok... so in order to tell if it was uniqiue one would have to do RREF, correct?