# 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

2

x+ 3y= 5

x− 3y= 7

Answer

We draw the 2 lines as follows.

2*x* + 3*y* = 5 is in green.

*x* − 3*y* = 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:

6

x− 3y= −12−2

x+y= 4

Answer

Once again, we graph the 2 lines and the intersection point gives the solution for the simultaneous equations.

6*x* − 3*y* = −12 has *x*-intercept `-2`,
and
*y*-intercept `4`.

−2*x *+* 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:

2

x− 3y= −6

x+y= 7

Answer

Once again, we graph the 2 lines and the intersection point gives the solution for the simultaneous equations.

2*x* − 3*y* = −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* − 5*y* = −10 has *x*-intercept `-10`,
and
*y*-intercept `2`.

*x *− 5*y *= 7 has *x*-intercept `7`
and has *y*-intercept `-7/5=-1.4`.

The graph is as follows:

Graph of the linear equations *x* − 5*y* = −10 and
*x *−* 5y *= 7 .

We see there are **no solutions** for the system since the lines are parallel.