Explain how and after how many steps you recognize a system of linear equation has no solution.

Question

Explain how and after how many steps you
recognize a system of linear equation has infinitely many solutions.

Solvimg systems of equations by elimination

Answer 1

In a worst case scenario the number of steps is equal to the number of equations, which should also be equal to to the number of unknown variables for which you are solving. 

For example, in the set

aX + bY + cZ = d

eX + fY + gZ = h

iX + jY + kZ = m

X, Y, and Z are the unknown variables.   There are 3 of them so you need not fewer than 3 equations to solve the system.   In the worst case, it will take 3 steps before you realize that there is either no solution or multiple solutions to the system.    If there had been 4 equations, the worst case would be 4 steps, and so on and so forth.

You know you have encountered such a situation when the results of a simplification yield a logical fault, such as 4=7.  

Answer 2

It really depends on how complex the problem is as to how many steps it will take.  Eventually you will simplify the problem down to something like this:

3x-9 = 3x+4  so -9 = 4  ???

If you get that result that means that there is no possible solution since no value you substitute for x will make the statement true.

You may also get something that simplifies down to this:

3x + 4 = 3x +4  so 4 = 4

If you get this result it means you have infinite solutions since any value you substitute for x will make the statement true.

 

Hope this helps for the problem you are working on