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Simultaneous Linear Equations

Geometric or Graphical Interpretation

The graph of a linear equation  ax+by = c  is a straight line.

Two distinct lines always intersect at exactly one point unless they are parallel (have the same slope).

The coordinates of the intersection point of the lines is the solution to the simultaneous linear equations describing the lines. So we would normally expect a pair of simultaneous equations to have just one solution.

Let's look at an example graphically:

2x + 3y = 7

4x + y = 9

From the graph we see that the point of intersection of the two lines is (2, 1)

Hence, the solution of the simultaneous equations is x = 2, y =1.

If you solved the two equations using either Gaussian elimination or substitution, you would obtain the same result.

Recall that the slope of the line  ax+by = c  is (b0).
Two lines are parallel if they have the same slope.
Thus if  a₁x + b₁y = c₁  and   a₂x + b₂y = c₂ are parallel lines then    .
And           

So, if the above equation is true, the lines are parallel, they do not intersect, and the system of linear equations has no solution.

Consider the following system of linear equations:

          x - y = -2

          x - y = 1.

Using the method of substitution, we subtract the second equation from the first to obtain: 0 = -3. This is a false statement and the system, therefore, has no solution.

If we look closer at the lines we see that they satisfy the condition

and are therefore parallel (as can be seen below). They do not intersect explaining why the system of linear equations has no solution.

What if the two equations represent the same line?

Consider the equations

x - y = 1

2x - 2y = 2

Multiply the first equation by 2 to put the equations in the form

2x - 2y = 2

2x - 2y = 2

Now subtraction gives 0 = 0, which is true no matter what values x and y may have! This time the two equations represent the same line, since both can be written in the form y = x - 1.

Any point on this line has coordinates which will satisfy both equations, so there are an infinite number of solutions!

In general, two equations represent the same line if one equation is a multiple of the other. That is          

There are then three possibilities for a pair of simultaneous linear equations:

(i) Just one solution (the usual situation - both lines are unique and not parallel to each other)

(ii) No solution ( the lines are parallel, )

(iii) Infinitely many solutions (the equations represent the same line, )

Exercise 4.

Determine whether the given pair of linear equations has a unique solution, no solution or infinite solutions.

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