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How do I determine if there is a solution to the following simultaneous equation?:

2a+3b=7

5a+7b=19

9a+5b=32

I have given a specific example but I was interested in the general case. I am confused because I have seen examples with two equations and two unkowns where it has been argued that, if, for example, I have:

2a+4b=x_{1}

5a+6b=x_{2}

Then:

[tex]a=\frac{1}{2}(x_{2}-\frac{3}{2}x_{1})[/tex]

[tex]b=\frac{1}{4}({\frac{5}{2}x_{1}-x_{2})[/tex]

and I can plug in any x_{1}and any x_{2}and get an answer, as there is no division by zero, etc, and so there is no reason why I can't get an output for a and c.

I can represent my original three equations in a similar format; presenting a as a combination of the outputs and b as a combination of the outputs. But when I put them in to the original equation it doesn't work!

Has this something to do with an assumption being made? That with two equations and two unknowns, I can always find a solution for any x_{1}and x_{2}(provided that they're not parallel), but for three equations and two unknowns, there is not always going to be a solution? If this is the case, how do I ascertain that there is not a solution (without graphing the functions)?

As always, any help appreciated.

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# Homework Help: Three equations, two unknowns

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