8.2 Systems of Three Equations and Three Unknowns
8.1 Systems of Two Equations and Two Un-
knowns
Here you will review how to solve a system of two equations and two unknowns using the elimination method.
The cost of two cell phone plans can be written as a system of equations based on the number of minutes used and
the base monthly rate. As a consumer, it would be useful to know when the two plans cost the same and when is
one plan cheaper.
Plan A costs $40 per month plus $0.10 for each minute of talk time.
Plan B costs $25 per month plus $0.50 for each minute of talk time.
Plan B has a lower starting cost, but since it costs more per minute, it may not be the right plan for someone who
likes to spend a lot of time on the phone. When do the two plans cost the same amount?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/61405http://www.youtube.com/watch?v=ova8GSmPV4o James Sousa: Solving Systems of Equations by Elimination
Guidance
There are many ways to solve a system that you have learned in the past including substitution and graphical
intersection. Here you will focus on solving using elimination because the knowledge and skills used will transfer
directly into using matrices.
When solving a system, the first thing to do is to count the number of variables that are missing and the number of
equations. The number of variables needs to be the same or fewer than the number of equations. Two equations
and two variables can be solved, but one equation with two variables cannot.
Get into the habit of always writing systems in standard form:Ax+By=C. This will help variables line up, avoid
+/- errors and lay the groundwork for using matrices. Once two equations with two variables are in standard form,
decide which variable you want to eliminate, scale each equation as necessary by multiplying through by constants
and then add the equations together. This procedure should reduce both the number of equations and the number of
variables leaving one equation and one variable. Solve and substitute to determine the value of the second variable.
Example A
Solve the following system of equations: 5x+ 12 y=72 and 3x− 2 y=18.
Solution: Here is a system of two equations and two variables in standard form. Notice that there is anxcolumn
and aycolumn on the left hand side and a constant column on the right hand side when you rewrite the equations as
shown. Also notice that if you add the system as written no variable will be eliminated.
Equation 1: 5x+ 12 y= 72