Solving Systems of Linear Equations

Date: 02/09/99 at 18:02:18
From: Taylor Kopacka
Subject: Solving systems of linear equations

Dear Dr. Math,

I am given two equations and one ordered pair. [y = 6x+12, 2x-y = 4  
(-4, -12).] How do I tell if the ordered pair is a solution of the 
system? Next, if I'm only given the 2 equations, how do I find the 
solution to graph? 

Thank you for all your help.

Taylor

Date: 02/10/99 at 12:54:20
From: Doctor Peterson
Subject: Re: Solving systems of linear equations

A solution to a system of equations is any set of values that satisfy 
all the equations in the system. So to see if a given ordered pair is 
a solution to the system, all you have to do is see if they are a 
solution of each individual equation. Plug in the values x = -4, 
y = -12 and you'll see that both equations are in fact true.

If you are told to graph the two equations and their solution, you can 
start by just graphing the two equations. You'll find two lines that 
may or may not intersect. (If they appear not to be parallel, but don't 
intersect within the range of values you have graphed, try graphing 
more values so they will.) You may well be able to guess the 
coordinates of the intersection from your graph, and just test it.

If that doesn't work (maybe the solution isn't a pair of integers), 
you'll have to use one of several methods to solve the equations 
algebraically. I don't know what you've learned in that area, but the 
simplest in this case is just to write both equations in the form 
y = mx + b, and set the two expressions for y equal to one another. 
In your case,
   
   y = 6x + 12   and   2x - y = 4

become

   6x + 12 = 2x - 4,

which we can solve by subtracting 2x from both sides and subtracting 
12 from both sides:

   6x + 12 = 2x - 4
   4x + 12 = -4
        4x = -16
         x = -4

From that you can get the value of y from either of the two equations 
and you're done.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/