224
6.2 Solving Systems Graphically
Now that we know how to solve complicated equations, we move on to solving what are called systems of
equations. A system of equations is when we have multiple equations with multiple variables and we are looking
for values that the variables represent so that all of the equations are true at the same time.
We will mainly be dealing with two variables and two equations, but you can solve most systems of
equations as long as you have the same number of equations as variables. As a quick example, consider the
following system:
+ 1 = 5
− 1 = 1
It doesn’t take too much work to verify the solution of this system is = 3 and 1 = 2. Notice that those
values for and 1 make both equations true at the same time.
3 +2 = 5
3 −2 = 1
The question remains, how do we get that solution?
Solving with Graphs
If we have our equations set up using the and 1 variables, we can graph both equations. Let’s see how
this helps us. To start with, let’s graph the first equation + 1 = 5. Remember that we can do this in a couple of
ways. We could simply make an /1 chart and plot the points. Alternately, we could get the equation in slope-
intercept form and then graph.
Let’s start with an /1 chart. Remember that in an /1 chart we pick values and substitute those into
the equation to find 1 values. Confirm on your own that this /1 chart is correct for .1=5:
-
2
-
1
0
1
2
1
7
6
5
4
3
Now we can plot those points on a coordinate plane and connect them to get our graph.
If we don’t like the /1 chart method, we can turn the equation into slope-intercept form by isolating the
1 variable on the left side like so:
.1=5
− + 1 = 5 −
1 = 5 +(−)
1 = − + 5
Subtract
from both sides
Subtract means add a negative
Commutative property
