How many solutions does the following system have:

Question

2x + 5y = 10
3x + 6y = 12
Explain how to determine the number of solutions without solving the system. Then apply elimination, and interpret the resulting equation.

Answer 1

The key is to calculate Δ, required for the first part of Kramer's rule.
Δ=determinant of the left-hand side.

If Δ≠0, then there is a unique solution, including the trivial solution of x=0 and y=0 if the right-hand side is all zeroes.

If Δ=0, then the left-hand side of the equations are linearly dependent.
Two cases may arise:
1. If the equations are consistent, one single equation will result after reduction of the left-hand side.
This case has infinitely many solutions, one set for each value we assign to x (or y).
2. If the equations are not consistent, then after reduction, the equations will be identical on the left-hand side, but the right-hand sides will different.

This gives a total of 3 different cases.