Posted by Kathy camkat on Saturday, March 5, 2011 at 10:49am.
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.

math  MathMate, Saturday, March 5, 2011 at 8:01pm
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.
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