Geometric or Graphical Interpretation
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
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
Multiply the first equation by 2 to put the equations in the form
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, )
Determine whether the given pair of linear equations has a unique solution, no solution or infinite solutions.