3. Graphical Solution of a System of Linear Equations
A `2 ×2` system of equations is a set of 2 equations in 2 unknowns which must be solved simultaneously (together) so that the solutions are true in both equations.
We can solve such a system of equations graphically. That is, we draw the graph of the 2 lines and see where the lines intersect. The intersection point gives us the solution.
Example 1
Solve graphically the set of equations
2x + 3y = 5
x − 3y = 7
Answer
We draw the 2 lines as follows.
2x + 3y = 5 is in green.
x − 3y = 7 is in magenta.
Graphs of `y = (-2x-5)/3` and `y=(x+7)/3`.
We observe that the point (4,−1) is on both lines on the graph. We say (4,−1) is the solution for the set of simultaneous equations.
This means the solutions are `x = 4`, `y =-1`.
Notice that these values are true in both equations, as follows.
2(4) + 3(−1) = 8 − 3 = 5 [OK]
(4) − 3(−1) = 4 + 3 = 7 [OK]
So we see the intersection point of the 2 lines does give us the solution for the system.
Types of solutions
A `2 ×2` system of linear equations can have three possible solutions.
1. Intersect at a point, so one solution only
Graph of the linear equations `y = x+3` and `y = -2x+13`.
2. Are parallel, so no intersection
Graph of the linear equations `y = -x+3` and `y = -x+7`.
3. Are identical, so intersect everywhere on the line
Graph of the linear equations `x+y = 6` and `2x+2y = 12`.
Example 2
Solve graphically the system:
6x − 3y = −12
−2x + y = 4
Answer
Once again, we graph the 2 lines and the intersection point gives the solution for the simultaneous equations.
6x − 3y = −12 has x-intercept `-2`, and y-intercept `4`.
−2x + y = 4 has x-intercept `-2`, and y-intercept `4`.
The graph is as follows:
Graph of the linear equations `6x-3y=-12` and `-2x+y=4`.
We see the lines are identical. So the solution for the system (from the graph) is:
"all values of (x, y) on the line `2x-y=-4`".
(We normally write equations in normal form with a positive infront of the x term.)
Example 3
Solve graphically the system:
2x − 3y = −6
x + y = 7
Answer
Once again, we graph the 2 lines and the intersection point gives the solution for the simultaneous equations.
2x − 3y = −6 has x-intercept `-3`, and y-intercept `2`.
x + y = 7 has x-intercept `7` and has y-intercept `7`.
The graph is as follows:
Graphs of `y = (2x+6)/3` and `y=-x+7`.
So we see there is one solution for the system (from the graph), and it is `(3, 4)`.
Example 4
Solve graphically the system:
x − 5y = −10
x − 5y = 7
Answer
For this system, we have:
x − 5y = −10 has x-intercept `-10`, and y-intercept `2`.
x − 5y = 7 has x-intercept `7` and has y-intercept `-7/5=-1.4`.
The graph is as follows:
Graph of the linear equations x − 5y = −10 and x − 5y = 7 .
We see there are no solutions for the system since the lines are parallel.