Simultaneous Equations Diet Balancing

Simultaneous Equations Used to balance for 2 nutrients Usually protein and energy Balances for required amounts Lb, kg, or Mcal Uses 2 ingredients or 2 mixtures of ingredients

Simultaneous Equations Identify feeds and their composition Feed A – 10% CP, 80% TDN Feed B – 35% CP, 50% TDN

Simultaneous Equations Identify animal requirements 0.8 lb CP and 4.0 lb TDN per day Let A = lb of Feed A needed Let B = lb of Feed B needed

Simultaneous Equations Lb Feed A Lb CP reqmt Equation describing CP Diet must supply 0.8 lb of CP so (A x 0.10) + (B x 0.35) = 0.8 lb CP Lb Feed B CP in Feed ACP in Feed B

Simultaneous Equations Lb TDN reqmt Equation describing TDN Diet must supply 4.0 lb of TDN so (A x 0.80) + (B x 0.50) = 4.0 lb TDN Lb Feed ALb Feed B TDN in Feed ATDN in Feed B

Simultaneous Equations 2 equations with the same unknowns 0.10A B = 0.8 lb CP 0.80A B = 4.0 lb TDN

Simultaneous Equations Multiply 1 equation by a factor that will make one unknown the same in both 0.10A B = 0.8 lb CP 0.80A B = 4.0 lb TDN 0.80/0.10 = 8, so multiply 1 st eq by 8

Simultaneous Equations 8 (0.10A B) = 8 (0.8 lb CP) 0.8A + 2.8B = 6.4 lb new equation

Simultaneous Equations Subtract the 2 nd eq from new equation 0.8A + 2.8B = 6.4 lb new equation 0.8A + 0.5B = 4.0 lb 2 nd equation B = 2.4

Simultaneous Equations Solve for unknown B = 2.4 B = 2.4/2.3 B = 1.04 What was B? Lb Feed B

Simultaneous Equations Substitute 1.04 for B in either original eq 0.10A B = 0.8 lb 0.10A (1.04) = A = 0.8 – A = A = 0.436/0.10 = 4.36 lb Feed A

Simultaneous Equations Our diet consists of: 1.04 lb Feed B 4.36 lb Feed A

Simultaneous Equations Check our work: 1.04 lb Feed B x 0.35 = lb CP 4.36 lb Feed A x 0.10 = lb CP 0.8 lb CP

Simultaneous Equations Check our work: 1.04 lb Feed B x 0.50 = 0.52 lb TDN 4.36 lb Feed A x 0.80 = 3.48 lb TDN 4.0 lb TDN