How Can You Derive the Sine of Alpha from Two Inclined Plane Equations?

[Kam Jam]

Homework Statement

"Combining the x-direction conditions in Equations 1 and 2, please show that $$sin ∝ = \frac {m_1 + m_2} {2M}$$"

The two equations below are describing two different setups of an inclined-plane system with a block of mass M attached to a hanging mass of ##m_1## in setup 1 and a higher ##m_2## in setup 2.

Homework Equations

Equation 1 = $$m_1g + f - Mgsin∝ = 0$$
Equation 2 = $$m_2g - f - Mgsin∝ = 0$$

The Attempt at a Solution

For a similar problem, I was able to set the two equations equal to one another and isolate the needed variable. I attempted to do the same, like so $$m_1g + f - Mgsin∝ = m_2g -f - Mgsin∝$$
With this method, I couldn't find any algebraic method which would allow me to add the masses; in order to put ##m_1## and ##m_2## together in any way, I'd have to subtract one from the other. I similarly couldn't cancel ##f##. The best I could do resulted in either ##2f## or ##-2f##.
I'd appreciate any help or guidance to a better solution.

 

[haruspex]

What variable is conspicuous by its absence from the target equation?

 

[Chandra Prayaga]

Could you please make a diagram? If there are two situations please draw them both, and identify the variables.

 

[Jhenxz]

That's a 2x2 Linear Equation System. Try another method to solve it!

Hint: You don't need the f, what is the best method to remove it?

 

[kuruman]

... I'd have to subtract one from the other ...

How about adding one to the other instead of subtracting?