The size (or dimension) of a matrix is expressed by writing: "number of rows" x "number of columns."
For example, if a matrix has 3 rows and 2 columns, then we say that the matrix is a 3 x 2 matrix.
Each equation corresponds to a row in the augmented matrix.
Each variable corresponds to a column to the left of the vertical line in the augmented matrix.
The constant term for each equation goes in the last column (to the right of the vertical line in the augmented matrix).
There are three row operations! We are allowed to do the following.
Multiply a row by a nonzero constant.
Switch two rows.
Add a nonzero multiple of one row to another row.
We perform row operations (see the question above for what row operations are allowed) to get the matrix into the following form.
The pivot element becomes a 1.
The other entries in that column become 0's.
In other words, we transform the matrix using row operations so that the column containing the pivot entry is a unit column with a 1 in the location of the pivot entry.
The following should be included in your work in order to get full credit on application problems.
Define the variables clearly. (This means including a quantifier. For example, don't just say x = corn and y = strawberries. Be specific about what the variable describes. Instead, you could say x = pounds of corn harvested and y = pounds of strawberries harvested.)
Write equations. (Look at the details and constraints of the problem to write equations.)
Solve the system of equations.
Answer the question (and be sure to include units.)