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There are two main approaches to resolve this issue:
- See if the determinant is zero. This will tell you if there is a unique solution, or if there are zero or an infinite number of solutions. If it's not unique, it won't tell you if it's zero or an infinity.
- Reason geometrically, as follows:
For a system with two equations, two unknowns, remember you can think of each equation as a line in an x-y coordinate system.
One solution means the lines cross.
No solutions means they are parallel.
An infinite number of solutions means they are the same line.
- Two lines cross if they don't have the same slope.
- Two lines are parallel if they have the same slope and a different y-intercept.
- Two lines are identical if they have the same slope and the same y-intercept.
Looking at the equations above, you see that they both have the same slope (2), but different y-intercepts (1, and -3). This means you have two parallel lines, and zero solutions.