Question

How can you determine the best method of solving a system of equations?

Answer 1

Your most common method is probably "elimination". That is where you manipulate
the equation so either the x's or the y's have the same coefficients, or opposite coefficients.
If one of the equations contains a variable with a coefficient of + or -1, then solve for
that variable and use "substitution".

e.g. (elimination)
3x + 2y = 8 --- multiply by 3 --> 9x + 6y = 24
4x + 3y = 9 --- multiply by 2 --> 8x + 6y = 18
notice I got the y's to be the same, so subtract them
x = 6
now sub that back into either of the original equations to find y

substitution:
4x + y = 9
3x - 2y = 4
from the first: y = 9-4x
into the 2nd:
3x - 2(9-4x) = 4
3x - 18 + 8x = 4
11x = 22
x = 2
in y = 9-4x
y = 9 - 4(2) = 1

After doing several thousand of these, like I have done in my lifetime,
it becomes easy to decide. But often it doesn't matter, since you will end
up with the same answer no matter what method you use.

If you had a case such as
y = 7x + 14
y = -3x - 26
you could use a third method called "comparison" where you simple equate
the value of the same variable:
7x + 14 = -3x - 26
10x = -40
x = -4, then in either of the origianls,
y = 7(-4) + 14 = -14