math
posted by Kathy camkat .
How many solutions does the following system have:
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.

The key is to calculate Δ, required for the first part of Kramer's rule.
Δ=determinant of the lefthand side.
If Δ≠0, then there is a unique solution, including the trivial solution of x=0 and y=0 if the righthand side is all zeroes.
If Δ=0, then the lefthand 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 lefthand 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 lefthand side, but the righthand sides will different.
This gives a total of 3 different cases.