# System of Equations with Two Unknowns, Algebra Help

1. Dec 31, 2013

### kwixson

1. The problem statement, all variables and given/known data

I have two equations with two unknowns. I know m, F, R, r and I. I need to find a and f.

m = 2
F = 28
R = 0.25
I = 0.0625
r = 0.1875

I know the ultimate answers are $a = 16\frac{1}{3}$ and $f = 4\frac{2}{3}$

2. Relevant equations

(1) $Fr-fR=I\frac{a}{R}$

(2) F+f=ma

3. The attempt at a solution

$Fr-I\frac{a}{R}=fR$

$f=\frac{Fr}{R}-I\frac{a}{R^2}$

$I(1+\frac{r}{R})=a(m+\frac{I}{R^2})$

$a=F(\frac{1+\frac{r}{R}}{m+\frac{I}{R^2}})$

$28(\frac{1+\frac{0.1875}{0.25}}{2+\frac{0.0625}{0.25^2}})$ = ?

16.333 (Yay!)

I can't get back to f from here. :( Help!

Last edited: Dec 31, 2013
2. Dec 31, 2013

### LCKurtz

From (2) $f = ma - F$ so plug your numbers in.

3. Jan 1, 2014

### kwixson

Ok, fair enough. But the answer guide said you could arrive at the following equation for f and I want to know how they got it:

$f=\frac{\frac{r}{R}-\frac{I}{mR^2}}{1+\frac{I}{mR^2}}$

4. Jan 1, 2014

### ehild

F is missing from the expression for f. It should be

$f=F\frac{\frac{r}{R}-\frac{I}{mR^2}}{1+\frac{I}{mR^2}}$

Plug in your equation $a=F(\frac{1+\frac{r}{R}}{m+\frac{I}{R^2}})$ for a into the equation f+F=ma.

ehild

5. Jan 1, 2014

### vela

Staff Emeritus
In matrix form, your system of equations is
$$\begin{pmatrix} R & \frac{I}{R} \\ -1 & m \end{pmatrix}\begin{pmatrix} f \\ a \end{pmatrix} = \begin{pmatrix} Fr \\ F \end{pmatrix}$$ A good technique in cases like this one is to use Cramer's rule. It'll get you the solution with minimal algebra.

http://en.wikipedia.org/wiki/Cramer's_rule#Applications