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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

Answer

We draw the 2 lines as follows.

2x + 3y = 5 is in green.

x − 3y = 7 is in magenta.

1 2 3 4 5 6 7 8 9 -1 -2 -3 -4 1 2 3 -1 -2 -3 -4 -5 x y
`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

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:

2 4 6 8 -2 -4 2 4 6 8 x y
`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

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:

1 2 3 4 5 6 7 8 9 -1 -2 -3 -4 1 2 3 4 5 6 7 8 9 10 -1 -2 x y (3,4)
`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

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:

2 4 6 8 -2 -4 -6 -8 -10 2 4 6 -2 -4 x y
`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.

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