Expert: Josh - 12/13/2005

Question
3p - q =10
2p - q =7
I know the answer. p=3 and q= -1
My question is: Is graphing the only way to solve
this problem. If not, what is the other way? Please explain.  Thank you!

Answer
Hi Ellen,

Good question. Graphing is only an ad-hoc method for solving systems of equation. It is often first taught in school because it is easy to grasp and people can visualize what's going on (when the equations are simple to plot). The solution(s) has a nice interpretation as being the intersection of curves.

In general, the solutions are found using algebraic techniques. In high school, you will probably proceed as follows:

3p-q=10 ...call this [#1]
2p-q=7  ...call this [#2]

multiply [#1] by 2 to get 6p-2q=20 ...call this [#1a]
multiply [#2] by 3 to get 6p-3q=21 ...call this [#2a]

Idea: eliminate one variable, say, "p" to find out answer to "q", then back-substitute value of "q" into either equation [#1] or [#2] to solve for "p"

subtracting [#1a] from [#2a], we get
-q=1, or q=-1.
Putting this back into [#1], you get 3p-(-1)=10,
3p=9, p=3.
Solutions: p=3,q=-1.

The problem may also be set up using matrices, that is, you abbreviate the notations and only consider the coefficients. This is easy to explain on paper, but extremely difficult to type here in this format because things do not align.