How can I determine if a system of equations has one solution, no solutions or many solutions?

Question

My question is self explanatory... Please be as throughrough as possible because I have to determine that in many systems of equations...

Answer 1

systems are solved for solutions ...
these solutions have 3 possibilities ....

1) one solution .... (where two lines intersect at one particular point .... the slope of the two lines will be diffreent)

2) no solution ..( where the two lines are parallel ... the equations will have the same slope, but never intersect)

3) infinite solutions... ( where one eq. is a scalar multiple of the other.... they are the same equations and therefore have infinite solutions)


examples ....
1)...3x + 2y = 12 ..... 3x - 2y = 12
( the negative sign makes the two eq.'s have different slopes ... therefore ..1 solution

2) 3x + 2y = 12 .......... 6x + 4y = 36
(here, the eq.'s have the same slope, but never intersect... no solution)

3) 3x + 2y = 12 .......... 9x + 6y = 36
(here, eq. #2 is a scalar of eq 1 ...[ eq2 = 3{eq1)] this makes every solution to 1 is a sol. to 2 ... infinite solutions ...

hope this explains your question

Answer 2

System Of Equations Infinite Solutions

Answer 3

Hi,
Since you have a number of equations, I strongly suggest that you use a graphing calculator if you have one. Just enter the equations as matrices and solve. If you have no solutions, one of the rows will have a false answer such as 8=0; if there are many solutions, you'll get something like 0=0 in one of the rows; if there is one solution, you'll get an answer such as this for a system of three equations in three variables:
[[1 0 0 3]
[0 1 0 2]
[0 0 1 5]]

Incidentally, you do this using the rref function of the calculator. If you don't know how to do this, visit my Website at this URL:
www.angelfire.com/pro/fkizer

If you don't have a calculator try this:
1) For many solutions see if the equations are multiples of one another,., i.e., divide by some constant.

2) For the other types, it's probably faster to graph the equations. If the lines are parallel and separated, there no solution, if they cross there is a single solution, if they coincide there are many solutions. Assuming the equations are linear, you only need two points for each equation for a quick check.

Hope this helps.

FE.

Answer 4

By doing row reduction (that is, Gaussian elimination). You can do this using matrices of the coefficients and right hand sides. If your system is Ax = b, where A is a matrix and b is a vector with the right -hand side values, it is easy to do row reduction using the matrix [A b]

Terminology: the first nonzero value in each row of A, during row reduction, is called a pivot.

Now, the answer to your question. If, after doing row reduction, there is a pivot in each row of A, then there is a solution (but it is not necessarily unique). If there is a pivot in each column of A, then any solution will be unique (but it might not exist). A unique solution exists if and only if there is a pivot in each row and each column.

Answer 5

15x-3x=12 y=5x-4