(Based on Section 2.2 in Finite Mathematics and Finite Mathematics and Applied Calculus)
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Solution of Systems of Equations by Row Operations, Continued
The following discussion is based on Section 2.2 of Finite Mathematics and Finite Mathematics and Applied Calculus.
Definition of RowReduced Echelon Form A matrix is in rowreduced echelon form (or reduced for short) if:

Q Select which (if any) of the following matrices are in rowreduced echelon form. 
Q What is so interesting about rowreduced echelon form?
A Recall the procedure you used in Part B to solve systems of linear equations. (If you don't, go back to that part of the tutorial.)
You used pivoting to clear the columns containing the leading entries, and then, as a final step, you converted the leading entries into 1's. Thus, what you were left with (apart from a possible rearrangement of the rows) was a rowreduced echelon matrix! In other words,
Q Decide whether the given matrix is in rowreduced echelon form. If it is not, finish the reduction.
Q Now that we know how to reduce a matrix to rowreduced echelon form, what do we do with it?
A First, look at Choice C directly above. You know by now that it is in rowreduced form. Also, you know from Part B of this tutorial that it happens to represent the following solution to a system of two linear equations in two unknowns.
Consider the system
Q How do we read off the solution from the reduces form. We never had to deal with a row of zeros before!
A Since we have done all we can with the matrix, we translate the rows of the reduced matrix back into equations, and we get:
x    3  =  3  . 
The second row tells us absolutely nothing: that 0 = 0. In other words, we are left with only one equation in two unknowns. Since we are left with only one equation in two unknowns, we have infinitely many solutions. To see what these solutions look like, solve the above equation for x, and write:
x  =  3  +  3 
This is called the general solution. Each choice of y will result in a different particular solution. Thus, for instance, if you choose y = 100, you get the particular solution
x  =  3  +  3  =  3 
y  =  100. 
Q The general solution of the system whose augmented matrix rowreduces to
3  
Q The general solution of the system whose augmented matrix rowreduces to
1  
Q The general solution of the system whose augmented matrix rowreduces to
1  
Q That is all very well. But what about nonstandard forms for larger matrices, such as
1  
2  ?  
0 
A Since rowreduced echelon form is as far as we can go with the matrix, we translate back into equations and obtain
x    5z  =  1  
y  +  9z  =  2  . 
x  =  5z1 
y  =  9z + 2. 
We can now choose z to be any number, and then get corresponding values for x and y according to the formulas, giving infinitely many solutions. Thus, the general solution is
x  =  5z1 
y  =  9z + 2. 
z arbitrary. 
We get particular solutions by choosing specific values for z. For example, z = 2 gives the particular solution
The following summary is adapted from Section 2.2 of Finite Mathematics and Finite Mathematics and Applied Calculus (Also, you can press the "Chapter Summary" button on the left to bring up a page with this and other information.)
Solutions of Systems of Linear Equations
One of three things will happen in any system of linear equations. There will be:
Hints

Now try the rest of the exercises in Section 2.2 of Finite Mathematics and Finite Mathematics and Applied Calculus.