How do you determine whether a linear system has one solution, many solutions, or no solution when given 5x+ 4y= -18 and 2x+3y=-24?
1 Answer
These have one solution
Explanation:
One can answer this in several ways!
Method 1
We can try directly solving the system of equations
by multiplying the second one by
which in turn implies
Thus, there is only one solution
Method 2
A bit more sophisticated method involves the coefficient matrix
The determinant of this matrix is
Hence this matrix is non-singular and so
Carrying out the explicit calculation will lead to the same answer as the first method (however, since all we need here is whether the solution exists or is unique, this last step is unnecessary - the existence of
Method 3
The two equations can be represented graphically by two straight lines, and the question of whether solutions exist then boils down to whether the lines intersect.
Since the slope of the line representing the first equation is
Note
- an explicit solution is easy in this case because there are only two equations in two unknowns - larger systems of equations are difficult to handle explicitly.
- the geometric picture (method 3) is easy to use because the graphical version is two dimensional
- method 2 will involve more effort for larger systems , and moreover, it only works when there are equal numbers of unknowns and equations (leading to a square matrix).
- for larger systems, a more advanced method like Gauss- elimination would be preferred.