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

I don't believe there is any one answer to your question. It's not too difficult to show whether a system of linear equations has a solution, or an infinite number of them. There are certainly many different techniques that can be brought to bear on different types of equations, but there is no one technique that one can use for all types of equations to determine whether there is a solution, or if there is, whether the solution is unique.

a(.09)+b=12490.10

a(.09)+b=1249.10

ebaines is right... without a proper idea of what you're talking about, it's hard to explain.

Here's an example though:

an equation with infinite solutions

A + B = 5

... A and B could be ANYTHING, as long as the sum = 5.

an equation with NO solutions

A + B = -2, when both A and B are positive.

Quote:

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Originally Posted by albybaby689

a(.09)+b=1249.10

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Here you have one equation and 2 unknowns. It's easy enough to check whether there is an infinite number of solutions: simply rearrange as:

b = 129.01- a(.09)

From this you can see that you are free to choose any value for a, and you get a corresponding value for b. You basically have a function b(a) = 1249.01 - (.09)a, which would plot on a graph as a linear function with an infinite domain, meaning 'a' can take on any value from minus infinity to positive infinity. Hence there are an infinite number of solutions.

one of the answer and the evergreen method to find the solution and the number of solutions is to draw the GRAPH of the given equation. I don't have a link but you must check the method of drawing graphs, you can solve 90% of questions using graph only!

An example of infinite number of solutions