How do you know when an equation has infinitely many solutions? How do you know when an equation has no solution?

One way to tell an equation has no solutions is note when you solve it you get conflicting results.

i.e Solve for x: 3x+2=3x-9

We can tell from looking at this that it has no solution, but lets do it.

Subtract 3x from both sides:

2=-9

See? 2 is not equal to -9. Therefore, it has no solution.

Also, suppose we have \frac{3x}{x-2}=1+\frac{6}{x-1}

Solving this for x yields x=\frac{6}{0}

Division by 0 is a big no-no. So, no solutions.

For infinite solutions. How can we tell. Say we have something like the system:

3x+y=6
6x+2y=12

We can see from an observation that the bottom equation is just the first one multiplied by 2. So, it has infinite solutions.

If we solved it we would get something like 12=12.

Which means it has infinite solutions.

It would be correct to say the solution is the set of all ordered pairs that satisfy

3x+y=6.

When you have a system and there are more equations than variables, then you have infinite solutions... more than likely.

See above, we have one equation but two variables in 3x+y=6

Does that help any?