Section 3.1: Linear Equations In Two Variables
We now introduce two variables into our equations. They are quite different than what we have seen previously
with one variable equations.
§1 Standard Form And Ordered Pairs
The standard form of a linear equation is Ax  By  C, where A and B cannot be zero and A,B, and C are real
numbers. Examples of linear equations in standard form are 3x  2 y  12 and x  y  0 . Note that there are
two variables now, x and y. How do we describe the solution to a two-variable equation?
It gets a little tricky here if this is your first time seeing them. You need to remember that a solution is any value
that makes the equation true when you substitute it for the variable. Here, we need to find values for x and for y.
So you need values for each variable. For example, if we look at the example above, 3x  2 y  12, if we let x = 2
and y = 3 then we see that it becomes a true statement – namely 3(2)  2(3)  12 . We have simply replaced
the x variable with the number 2 and the y variable with the number 3, and when we complete the arithmetic
we see that it becomes a true statement.
We give solutions to linear equations as an ordered pair – (x,y). The x variable always comes first and the y
variable always comes second. Hence the solution above can be written as the ordered pair (2,3).
We can determine if an ordered pair is indeed a solution to an equation by simply plugging in the values and
seeing if a true statement occurs. For example, is the ordered pair (4,1) a solution to the equation
2 x  3 y  5? Well, by plugging in the 4 for x and the 1 for y and doing the arithmetic, we see that a true
statement does occur. That means that (4,1) is a solution! How about the ordered pairs  6,3 and 1, 1 ? We
see that the first ordered pair is not a solution but the second ordered pair is. This tells us something – for any
linear equation there are more than one solution – in fact, there are infinitely many!
PRACTICE
1) Determine whether the following ordered pairs are solutions to the equation 3x  y  6 : (2,0), (1,-3), (4,4)
and (3,3)
§2 Finding Solutions To Linear Equation
For the most part, you will be asked to find solutions to linear equations. For example, say we are given the
equation y  3x  1 and you are asked to complete the ordered pair (5,__). What are you being asked to find?
You are asked to find the value of y when x = 5. Hence we can plug in 5 for x and do the arithmetic to find y. We
see that when x = 5, y = 14. Hence the ordered pair is (5,14). Similarly, complete the ordered pair (__, 5). This is
asking you to find the value of x when y = 5. So if you plug in y = 5, we get that x = 2. Hence the ordered pair is
(2,5). Note that these are just two of the many solutions to the linear equation!
We can also make a table of values to give the solutions. The main concept is this – for any x value that you
pick ,there must be one corresponding y value, and vice versa. If you are given values of x or y, then plug them in
to find the ordered pair solution. However, if you are not given values of x or y, then you can actually go ahead
and pick any value that you want! Of course, it may be easiest to start with 0, 1 and 2.
For example, say we want to complete the following table for 2 x  y  5
x
3
y
0
-1
You can verify then that the solutions are (3,1), (5/2, 0), and (-1,7).
Similarly, if you were asked to find 3 ordered pair solutions but are not given any x or y values, you can go ahead
and pick any values that you want. Note that 0, 1, and 2 are the easiest to pick. So we can see that (0,-5), (1,-3)
and (2,-1) are also solutions!
PRACTICE
2) Complete the table of values for the equation 4 x  3 y  24 :
x
3
0
y
0
6
3) Find three ordered pair solutions for the equation 3x  2 y  18
§3 Plot Ordered Pairs
When we plot ordered pairs, the first thing to note is that there are two variables. Hence our graph must have
two variables as well. This is achieved through the rectangular coordinate system, aka the x-y plane. There are
two axis here – the x-axis and the y-axis. The point in the middle, also called the origin, is the point (0,0).
The horizontal number line is called the x-axis. The vertical line is called the y-axis. Whenever we plot a point on
the x-y plane, we can think of it as a grid – how many units away from the origin the point is. For example, if we
want to plot any ordered pair (x,y), we always plot x-coordinate first. If x is positive, we move to the right of the
origin and if x is negative we move to the left of the origin. Then we turn and move up or down the number of
units that correspond to the y-coordinate – we move up if y is positive and we move down if y is negative.
Say you are asked to plot the point (2,3) on the x-y plane. It would be as follows:
You need to remember that a negative x value means you go to the left and a negative y value means we go
down. Look at the following and verify the coordinates are as they should be:
PRACTICE
4) Plot the ordered pairs (4,0), (5,-2), (-6,-3), (0,-1) and (-4,3) on the same x-y plane.