Use Equations (1) and (2) to eliminate T and show that theexpression for f is gi

Question

Use Equations (1) and (2) to eliminate T and show that theexpression for f is given by:
f=g(m-Msin() - (m + M)a
Equation (1) T-Mgsin() - f = Ma Equation (2) mg-T = ma
------------------------- First I set the two equations equal to T:
1) => . T = Ma + Mgsin() + f 2) => . T = mg - ma
Then set (1) = (2)
. Ma + Mgsin() + f = mg - ma
Solve for f . f = mg - ma - Ma - Mgsin()
Simplify . f = m(g-a) - M(a-gsin()) ^^^ The answer I get is opposite from what the equation hasderived.
Can somebody check my work real quick?
Thanks

f=g(m-Msin() - (m + M)a
Equation (1) T-Mgsin() - f = Ma Equation (2) mg-T = ma
------------------------- First I set the two equations equal to T:
1) => . T = Ma + Mgsin() + f 2) => . T = mg - ma
Then set (1) = (2)
. Ma + Mgsin() + f = mg - ma
Solve for f . f = mg - ma - Ma - Mgsin()
Simplify . f = m(g-a) - M(a-gsin()) ^^^ The answer I get is opposite from what the equation hasderived.
Can somebody check my work real quick?
Thanks

Explanation / Answer

f=g(m-Msin() - (m + M)a
Equation (1) T-Mgsin() - f = Ma Equation (2) mg-T = ma

T = Ma + Mgsin() + f
T = mg - ma

Ma + Mgsin() + f = mg - ma
f = mg - ma - Ma - Mgsin()
f = mg - Mgsin() - ma - Ma
f = g(m-Msin()) - (m+M)a

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