Lecture 7- Equation Addition Method for 3x3 Linear Systems Example 1) 2 a + 5 b +1c=7 8 b +1c= 19 Notice this one is already an equation between b and c! 10 a + 10 b + 5c=5 2 a + 5 b +1c=7 (8 b +1c= 19) We multiply the 2nd equation by -1, then add that to the 1st equation, so essentially we are 2 a - 3 b+0c= -12 subtracting the second equation form the first one Oc is just 0 Now we have an equation between just a and b: great! 2 a - 3b = -12 8 b +1c= 19 Let's rearrange this in terms of c: 1c is just c 8 b+c= 19 Subtract 8b from both sides Here I managed to write c in terms of b, great! Let's rearrange this in terms of c also: add 3b to both sides c= 19 - 8b 2 a - 3b = -12 2 a = -12 + 3b Divide both sides by 2 a = -6 + 1.5b And here I have created an equation that describes a in terms of b also! Armed with these two new equations, I can find out everything if I can just find b. Now that we have a and c in terms of b, we can take any equation that uses any of these variables and we can replace EVERYTING with b's, which means it'll be just b’s and numbers, so we will be able to solve for b. Here's a rule of thumb for plugging into equations to find new information: If you use the same equa- tions that you got that information from, you probably won't learn anything new, because you've already extracted the information out of them. So, let's use the third equation, which we haven't touched yet, and hopefully has some new information for us! 10 + 10 b + 5 c = 5 c = 19 - 8b Here is our third equation, we haven't used it yet This tells us how to convert c to b a = -6 + 1.5b This tells us how to convert a to b 10 (-6 + 1.5b) + 10 b + 5 (19 - 8b) = 5 Plugging everything in, we have this new equation, only involv- ing b & numbers! -60 + 15b + 10b + 95 - 40b = 5 Distributive the coefficients 35 - 15b = 5 Combining like terms, b's together, numbers together -15b = -30 Subtract 35 from both sides b = 2 Divide both sides by -15 Now that we have b, we can plug back into the c = 19 - 8b and a = -6 + 1.5b c = 19 - 8b c = 19 - 8(2) c = 19 - 16 c = 3 a = -6 + 1.5b a = -6 + 1.5(2) a = -6 +3 a = -3 Final Answer: a = -3, b = 2, c = 3 And then we should check our work by plugging into the original equations! Example 2) This equation relates a and c This equation relates b and c 10 a + +6c =76 9 b+4 c = 75 6 a + 5 b +c= 55 -2(10 a + +6 c = 76) Notice that -2*6c=-12c 3(9 b + 4 c = 75) and that 3*4c= 12c, so the 12c and -12c will destroy each other! The goal here is to destroy the c from the equations -20a 12c = -152 Distribute the -2 27b + 12c = 225 Distribute the 3