T₁ is tension in rope connecting 10.0 kg and 30.0 kg

T₂ is tension in rope connecting 10.0 kg and 15.0 kg

Sum Forces = mass x acceleration

Green Object    m_green g Sin 30^o - T₂ = - m_green a

Purple Object   m_purple g Sin 30^o + T₂ - T₁ = - m_purple a

Red Object      m_red g - T₁ = m_reda

NOW SOLVE THE EQUATIONS SIMULTANEOUSLY

T₁ = m_redg - m_reda                                                               (Red Object solved for T₁)

m_purple g Sin 30^o + T₂ - m_red g + m_red a = - m_purple a        (T₁substituted into Purple Object)

T₂ = m_red g - m_purple a - m_red a - m_purple g Sin 30^o            (Previous Equation Solved for T₂)

m_green g Sin 30^o - m_red g + m_purple a + m_red a + m_purple g Sin 30^o = - m_green a  

                                                                                              (T₂ substituted into Green Object)

NOW GET ALL a TERMS ON LEFT SIDE AND ALL g TERMS ON THE RIGHT SIDE

m_purple a + m_red a + m_green a = m_red g - m_purple g Sin 30^o -  m_green g Sin 30^o

FACTOR OUT a

a(m_purple + m_red  + m_green) = m_red g - m_purple g Sin 30^o -  m_green g Sin 30^o 

     
        (m_red g - m_purple g Sin 30^o -  m_green g Sin 30^o )
a = ----------------------------------------------------------------
                   (m_purple + m_red  + m_green)

  
       ( 30.0 kg x 9.80 m/s² - 10.0 kg x 9.80 m/s² x Sin 30^o - 15.0 kg x 9.80 m/s² x Sin 30^o)
a = -------------------------------------------------------------------------------------------------------------- =  3.12 m/s²
                                               (10.0 kg  +  30.0 kg  +  15.0 kg)

  
a = =  3.12 m/s²

m_green g Sin 30^o - T₂ = - m_green a  

T₂  =  m_green a  +  m_green g Sin 30^o T₂  =  15.0 kg x 3.12 m/²  +  15.0 kg x 9.80 m/s² Sin 30^o

T₂ = 120. N