MHB How Can We Construct a Unique Linear System From Given Conditions?

[orgekas]

Construct a linear system determined by four numbers whose sum is 40, with the first three numbers adding up to 20 and the last three to 30.

a) Explain why this system has infinitely many solutions.
b) Add another condition on the numbers so that a unique solution can be found and then find this solution.

Anyone can help?
Thanks.

 

[Evgeny.Makarov]

Please see rule #11 http://mathhelpboards.com/rules/ (click on the Expand button at the top of the list).

What do you know about systems of linear equations?

 

[orgekas]

Ok sorry for not showing my work.

But I didn't have much to show. I know the answer to the first question but not to the second one.

I've constructed my equations as following

x1 + x2 +x3 + x4 =40

x1 + x2 + x3 = 20

x3 + x3 +X4 = 30

I don't know what to do from there.

 

[Evgeny.Makarov]

The system is correct except that the last equation should say $x_2+x_3+x_4=30$ and $x$'s should be all lowercase or all uppercase (preferably, the former).

Do you know about the matrix of a system of equations? Echelon normal form? Rank of a matrix? Rouché–Capelli theorem? Determinants?

Note that by subtracting the second equation from the first one, $x_4$ is uniquely determined. But what about $x_3$? Can you let it equal any number and still find $x_2$ and $x_1$?

 

[orgekas]

Yes I do know all that stuff. I was thinking of letting x3 = 10?

 

[Evgeny.Makarov]

I was thinking of letting x3 = 10?

My question was, Can you let $x_3$ equal any number and still find $x_2$ and $x_1$? And the answer is yes. Therefore, the system has infinite number of solutions.