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#### Solving For Two Variables:

A. There
are several different ways of solving two equations with two variables.
This page will focus on only two ways:

1.
Substitution Method

2.
Elimination Method

B.
Substitution Method

1. This
method works under the assumption that you can solve for one variable (in terms
of another) and plug this value into the other equation and solve.

Ex [1] If
2x + y = 5 and 3x + 2y = 9, then x = ___.

a. In this
problem, the 1^{st} equation has a single 'y' so we can solve for y and
get y = 5 - 2x. Now we can substitute this value into the 2^{nd}
equation.

b. Doing
so gives 3x + 2(5 - 2x) = 9. Now we have an equation with one variable so
we can solve for x. This yields: 3x + 10 - 4x = 9 or x = 1.

c. The
answer is 1. If the problem had wanted the y-value, you substitute x =1
for one of the equations and solve for x. 2(1) + y = 5. So y = 3.

C.
Elimination Method

1. To use
this method, you will have to "eliminate" one of the variables by
adding the equations together. Sometimes you have to multiply one equation
by a constant to get the same coefficient (but they must have opposite signs) so
they will cancel each other out.

Ex [1] If
3x + 2y = 4 and -6x + 3y = 6, then y = ___.

a. In this
equation, we can eliminate the x variable if we multiply the first equation by 2
and add it to the second equation.

b. So
multiplying the 1^{st} equation by 2 gives: 6x + 4y = 8. Now we
can add the two equations together and get 7y = 14 or y = 2.

c. The
answer is 2. If the question had asked for the x value we could plug y=2
into one of the equations and solve for x. 3x + 2(2) = 4 or 3x = 0 or x =
0.

D.
Sometimes the problem can be solved much easier if the 2^{nd} equation
is a multiple of the first.

Ex [1] If
2x - 3y = 4 then 6x - 9y = ______

a. This
problem is a little different from the ones above, but is much easier. If
you notice the 2^{nd} equation is 3 times the 1^{st}. That
means the answer is 3 x 4 or 12.

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