Systems of Linear Equations : 3 Unknowns (Echelon Method)

The problem is to find all the possible solutions to the following:

Eq 1: x + y = 2
Eq 2: y + z = 3
Eq 3: x + 2y + z = 5

I set up my matricies in the following:

1 1 0 2
0 1 1 3
1 2 1 5

operation 1: (-1*row 1 +row 3)

1 1 0 2
0 1 1 3
0 1 1 3

operation 2: (-1*row 2 +row 3)

1 1 0 2
0 1 1 3
0 0 0 0

operation 3: (-1*row 2 +row 1)

1 0 -1 -1
0 1 1 3
0 0 0 0

I am taking a optimization class, but I have never had linear algebra. From the review problems in the book, I have got this far, but I don't know what else I could do now operational wise that would make sense, and I don't know what this means to the solution. Can you help explain it to me?

See attached
1) Best Rentals charges a daily fee plus a mileage fee for renting its cars. Barney was charged $ 147 for 3 days and 300 miles, while Mary was charged $ 268 for 5 days and 600 miles. What does Best Rental charge per day and per mile?
Write an equation for the line. Use slope-intercept form, if possible.

Please help me solve the following system of three equations and discribe the methods that are being used to help me understand:
X + Y + Z = 6
2X - Y + 3Z = 8
3X - 2Y - Z = -17
Thank you!

1. Guassian Elimination is really just using the regular Elimination Method and applying the Triangular Form procedures to a set of linearequations, but without having to write down all the variables in the equations each time. This in itself simplifies the procedure and helps focus on the task. What is the ultimate goal of all

Nine total questions. please show work so I can understand and finish
1)A lumber yard has fixed costs of $1463.00 a day and variable costs of $1.00 per boardfootproduced. The company gets $2.40 per board-foot sold. How many board-feet mustbe produced daily to break even?
2)Use the echelon method to solve the system of th

Two angles are complementary of each other. Twice one angle is equal to the other angle plus the product of three and five.
A. Set up a system of linearequations to represent the two angles. (Hint: You will need two equations and two unknowns.)
B. Graph each of the equations on one rectangular coordinate system. (Hint: Y

1) Solve by the addition method.
3x + 2y = 14
3x - 2y = 10
2) Solve by the addition method
5x = 6y + 50
2y = 8 - 3x
3) Solve. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
4) Can't type fractions, so

4x+y=12 (1)
x-y=8(2)
4x+y=12
x-y=8
5x/5 = 20/5
X=4
4(4) =y - 12
16+y = 12
Y =4
I think this would be an example of elimination method for solving a system of equations however I am unsure how it would transfer to substitution method thus I am needing assistance. I am needing this to be illustrated if you wi