Sistem Persamaan dan Pertidaksamaan Linier Dua Variabel, dan Sistem Persamaan Linier Tiga Variabel: Menyelesaikan Sistem Persamaan Linear Dengan Menggunakan Grafik

The solution of a system of linear equations is the point that satisfies all the equations in the system. Graphically, a linear equation represents a straight line where the points on the line are the points that satisfy the linear equation. So, for a system of linear equations the solution will be the point that lies on all the lines i.e. satisfies all the equations. Therefore, it can be said that the solution of a system of linear equations is the point of intersection of all the lines in that system. For example consider a system of linear equations:

y = x + 4

y = -x + 2

By substitution, the solution is calculated as (x=3 and y=-1) which is also the point of intersection of the two lines, as shown in the following figure:

As two distinct straight lines cannot intersect each other more than once, it implies that a system of two or more distinct linear equations can have a maximum of one solution where all the equations are satisfied. Similarly, if two equations have the same slope then they represent two parallel lines. As, two parallel lines do not touch or intersect each other, such a system of linear equations has no solution.

The solution of a system of linear equations can be determined by plotting the lines and finding their point of intersections. Following examples illustrate the method of solving the systems of linear equations graphically:

Example 1:

Solve the system of linear equations:

y = 3x + 3

y = 2x + 4

To find the solution both the equations must be plotted on the same graph. The plot of both these equations will be as follows:

It can be observed from the above graph that the two lines intersect each other at the point where x=1 and y=6.

This can be verified by analytical methods as:

Verification:

y = 3x + 3

y = 2x + 4

So,

3x + 3 = 2x + 4

3x - 2x = 4 - 3

x = 1

Substituting in y = 3x + 3

y = 3(1) + 3 = 3 + 3

y = 6

So the solution is x=1 and y=6.

If the two lines do not intersect each other then they do not have a valid solution. For example, consider the following system of linear equations:

Example 2:

x + 3y = 7

3x + 9y = 4

These two equations are plotted in the following graph:

It is apparent from the figure that the lines are parallel and will never intersect each other hence the graphical method tells us that the solution does not exist. It can be verified analytically as:

Verification:

x + 3y = 7 => x = 7-3y

substituting in the other equation

3(7 - 3y) + 9y = 4

21 - 9y + 9y = 4

21 = 4 (not true)

Hence, the solution does not exist