How to tell how many solutions a linear system has
Once we’ve row-reduced a system, we can tell how many solutions it has before we solve for every variable. Start by checking whether the rows can all be true at the same time. Then see whether any variables are free to change.
Check the last column for a contradiction
A row with all zero coefficients and a nonzero constant is an equation like . If a row says that, the system has no solution.
For example, take , , and . Subtract the first and second equations from the third. The variable terms cancel, but the right side becomes .
Find the pivots if the system is consistent
If the system has no contradiction and every variable column has a pivot, there’s exactly one solution. The following reduced matrix has a pivot in all three variable columns.
The solution is .
Use a parameter when a variable is free
Now take the first example and change the third constant from to . The same row operation gives . There are pivots for and , but is free. Let and solve for the other two variables.
Every real value of the parameter gives a solution. That’s how we get infinitely many solutions.