Solving Matrices

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Sparky09

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Jun 4, 2011
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Hey there,
I'm working with matrices now and I understand how they are solving the equations in the examples but in the questions we have to answer they give three unknowns and only two equations and are asking me to solve the linear system by using row reduction and interpret the solution geographically. I'm hoping someone can point out where I'm going wrong?
The equations given to solve are:
4x-y+z=2
x+3y-2z=-1

I know to use row reduction you would put the equation into a matrices and then eliminate x (normally) which would then give the answer for y and then substitute that back into the first plane. However, in this equation if I do that it leaves z as an unknown as well so that doesn't work at least not the way they've taught us. In a matrices it would look like:

[4 -1 1|2
1 3 -2|-1]

Please forgive my brackets not sure how to enter the big ones on here or the divider line representing the equal sign but hopefully you get the idea. Can anyone point me in the right direction on how to solve this?
 


Hi. The way that you typed the augmented matrix is clear enough.

Do you realize that there are infinite solutions?

When there are less equations than variables, we call it an "underdetermined" system. In pre-calculus, such a system is called consistent and dependent. The infinite solutions to your system could be expressed using z as a parameter. In other words, formulas for calculating the x- and y-coordinates in terms of the z-coordinate comprise the solution:

x = some expression in z
y = some expression in z
z = z

where z is any Real number.

So, perhaps you already arrived at something like this.

Otherwise, please show us what you get, after applying some reducing row operations to your augmented matrix to obtain this form:

[ 1 0 ? | ? ]
[ 0 1 ? | ? ]

Or, continue on and try to finish, by following the following example.

Let's say we have this reduced result for some system of two equations in x,y,z:

[ 1 0 2 | 3 ]
[ 0 1 4 | 5 ]

Write down the two equations represented by this reduced, augmented matrix.

x + 2z = 3
y + 4z = 5

Solve the first for x in terms of z.

Solve the second for y in terms of z.

The solution, in this example, is:

x = 3 - 2z
y = 5 - 4z
z = z

where z can be any Real number.

As long as no more than one of the coefficients A, B, and C is zero, the graph of an equation Ax + By + Cz = D is a plane. You have a system with two such equations. Hence, the graph of the system will be two planes. There are infinite solutions, so what do you think the solution set of all valid ordered triplets (x, y, z) looks like graphically?

 
Okay, I think I've worked through it now. Took me awhile to understand why and where you were coming from but your answer was spot on! Thanks so much :) I finally GET IT!
 
Tried a lot of times but not get perfection in this. But seems very boring in beginning, But in the next phrase liked it "problem sums". Still somewhere i still need more practice






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