Solving systems of equations from dynamics

I am studying first year undergraduate physics and am having difficulty solving the systems of equations that emerge from dynamics problems.

If I have as many equations as unknowns, I've been told the system has the possibility of being solved. How do I know if the equations are sufficient to solve? [Should each equation contain each variable?]

I'm familiar with elimination and substitution. Are there some guidelines for the cleanest approaches to systems of equations?

Here is an example of a system I might have to solve for $F$, given $M$, $m$, $\mu_S$, $\theta$.

$$N\sin\theta−\mu_SN\cos\theta=ma$$ $$N\cos\theta+\mu_SN\sin\theta−mg=0$$ $$−N\sin\theta+\mu_SN\cos\theta+F = Ma$$

• I replaced the example. – jaynp Feb 24 '13 at 5:13

What you have been told, is false. The system $x+y=1,x+y=17$ has as many equations as unknowns, yet it has no solution.

The best approach to systems of linear equations is (some variation of) Gaussian elimination.

If the equations are not linear (and even if they are linear, but there are lots of them, with big coefficients), the best approach is to type them into some computer algebra package, like Maple.

• I edited my original statement to be more accurate. On physics exams, I am required to solve the systems by hand. – jaynp Feb 24 '13 at 4:01
• What does it mean, "has the possibility of being solved"? Also, you haven't addressed the point of whether the equations are linear. It makes a big difference. – Gerry Myerson Feb 24 '13 at 4:24
• They are almost always linear and almost always contain trigonometry functions. I am familiar with Gaussian elimination; how can I use it with these types of systems? – jaynp Feb 24 '13 at 5:07
• If they are linear, and if you are familiar with Gaussian elimination, then I don't see where you have a problem. Maybe you could give an example of a linear system where you have trouble with Gaussian elimination, and point out exactly where along the way the difficulty arises. Maybe even as a new question. – Gerry Myerson Feb 24 '13 at 11:14