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# 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

We draw the 2 lines as follows.

2x + 3y = 5 is in green.

x − 3y = 7 is in magenta.

x-3y=7
2x+3y=5

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

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:

6x-3y=-12
-2x+y=4

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

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:

2x - 3y = -6
x + y = 7

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

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:

x - 5y = -10
x - 5y = 7

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.