Math help please...linear inequalities?

Question

Here is the question. I am not looking for a quick free answer. I am truely stuck and would like it explained so I can learn how to do it. thanks

Not all systems of linear equations have a solution. Give an example of a system of linear equations that has no orderd pair satisfying it.

"linear inequalities" is the topic we're working on

Answer 1

Any 2 'linear' equations describe 2 lines in the x-y plane. And the point at which these 2 lines cross is called the 'solution' of the system because it is the one and only set of (x,y) values that work in *both* equations.

There are 2 ways that a system of 2 equations can't be solved and one of them (that they never cross because they are parallel) has already been mentioned.

The other way for them not to have 'a' solution is if the 2 equations describe the same line. In this case there are an infinite number of solutions, but none of them are 'unique.

Later on you will learn that systems of 3 equations describe 3 'planes' in 3-dimensional space and there will be a single, 'unique' solution if, and only if, those three planes all intersect in one common point. This same principle also extends into larger numbers of equations with more variables, but it's not so easy for most people to 'visualize' more than 3 dimensions. (Mathematicians do it all the time, but we're all 'bout half crazy anyway ☺)

HTH ☺

Doug

Answer 2

In your title, you mention inequalities, but in the actual question, you mention equations. Which is it? I will answer first assuming you mean "equations".

Two linear equations have no common solution of the lines (assuming 2 variables, say x and y) don't intersect. The only way for two lines to not intersect is for them to be paralell.

Thus find equations whose graphs are pralell lines and you are done. an example is

y = x and
y = x + 1

Similarly, two linear inequalities have no common solution if the region obtained by graphing the solutions of each don't intersect, so, in particular, we need for the boundary lines to not intersect. Thus we start with two lines which don't intersect and turn them into inequalities:

y = x------------> y < x
y = x + 1-------> y > x + 1

Not all choices of ways to make these equations into inequalities will work. You have to make sure you are choosing region below the bottom of the lower line for one inequality, and the region above the upper line for the other.

I hope this was comprehensible for you.

Answer 3

x+2y>0
x+2y<0

simple. since both are divided by the same line going to different directions, the would never have the same solution set

Answer 4

Sure; here's one.

2x + y > 1
4x + 2y < 0

Answer 5

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