Skip to main content

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.

Easy to understand math videos:
MathTutorDVD.com

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`.

Continues below

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.)

Please support IntMath!

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)`.

Get the Daily Math Tweet!
IntMath on Twitter

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.

Easy to understand math videos:
MathTutorDVD.com

top

Search IntMath, blog and Forum

Search IntMath

Online Algebra Solver

This algebra solver can solve a wide range of math problems.

Algebra Lessons on DVD

Math videos by MathTutorDVD.com

Easy to understand algebra lessons on DVD. See samples before you commit.

More info: Algebra videos

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!


See the Interactive Mathematics spam guarantee.