There are two ways to think about the problem. Let's consider a problem where you have 3 equations with 2 unknowns (like problems 5, 6). The solution to this system of equations is the intersection of these three lines. Randomly draw 3 lines on a piece of paper. It is possible that
a.) They don't have a common intersection (for example, they might all be parallel; or 2 out of 3 could be parallel; or none of them may be parallel but they don't have a common intersection point). No intersection = no solution. Thus if you create the augmented matrix and put into echelon form, you will see a pivot in the last column.
b.) The three lines could intersect in a single point. Thus you have a unique solution. Thus if you create the augmented matrix and put into echelon form, you will have a pivot in each column of the coefficient matrix (but not the last column of the augmented matrix). Note unique solution implies no free variables. To determine the point of intersection, put matrix in REF.
c.) The three lines could all be the same. I.e., all three equations represent the same line. I.e., all three equations are multiples of each other. The intersection is a line. There are an infinite number of points on a line. Thus there are an infinite number of solutions (and hence you have a free variable when using matrix form).
Thus the answer is "obvious" (see below for an example of what obvious means in mathematics). If you use matrix form, the augmented matrix will have 3 rows and 3 columns. After simplifying to echelon form, you should get a matrix where the last two rows contain only zeros. Thus you can use the first equation (or any of the equivalent equations) to solve for x in terms of y (where y is the free variable).
Recall from Wednesday's class that there are only 3 possible types of answers to a system of linear equations.
1.) no solution.
2.) unique solution
3.) infinite number of solutions.
Note that this makes sense both geometrically as well as algebraically (via matrix form).
Problems 7 - 10 all have to do with intersecting planes. In problems 7 - 10 you have 3 variables. Thus each linear equation represents a 2-dimensional plane in 3-dimensional space, R3. Thus in problems 7 - 10, you want to find the intersection of planes in R3.
For problems 7 and 8, you have 2 equations with 3 unknowns. Thus you have 2 planes in R3 and want to find their intersection.
It is possible that
a.) you have 2 different parallel planes. Thus they don't intersect. Thus there is no solution. This can be determined via the echelon form of the augmented matrix.
b.) The two planes intersect in a single line. Thus you will have one free variable and can solve for the other two variables in terms of the free variable.
c.) The two planes are the same. I.e., both equations represent the same plane. I.e., the equations are multiples of each other. The intersection is a plane which is 2-dimensional (and hence you will have TWO free variable).
In problems 9 and 10, you have 3 equations with 3 unknowns. Thus you have 3 planes in R3 and want to find their intersection. Consider the possibilities...
Sidenote: A linear equation with 4 variables represents a 3-dimensional hyperplane living in 4-dimensional space, R4. This case and higher-dimensional analogues is a bit harder to visualize (but people try using for example color or time for the 4th dimension. Many real-life applications have a lot of variables, so the harder to visualize cases are often more common in real life).
Laplace's Mecanique Celeste, an enormous five volume tome on just about everything you ever wanted to know about celestial mechanics, was first translated into the English language by Nathaniel Bowditch. Though others did it before him, Laplace was notorious for leaving length demonstrations to the reader, usually preceded with "C'est visible que.." (It is obvious that..). Bowditch meticulously filled in all the gaps, but before long he grew to dread those words, for he knew that when he saw them, he was in for a lengthy bit of derivation before what Laplace claimed was obvious was, in fact, obvious.