Solving Systems of EquationsDate: 10/30/96 at 11:52:48 From: sylvia hobson Subject: algebra I thought I knew how to do this, but guess I don't. Solve for a, b, and c in at least two ways: 2a+6b+c = 5 a+b-c = -1 a+8b+2c = 0 Please help. Date: 10/31/96 at 17:40:57 From: Doctor Tracy Subject: Re: algebra Sylvia, You are right that there are many approaches to solving this problem. They are really all equivalent mathematically, but everybody seems to prefer one approach or another. I will show you how to solve your problem using two techniques and then you should be able solve other problems like this one. First of all, let's label your equations: (I) 2a + 6b + c = 5 (II) a + b - c =-1 (III) a + 8b +2c = 0 Technique 1: The first thing that we want to do is eliminate one of the variables. Let's try to eliminate a. Notice that if you add -2 times the second equation to the first equation, the a's will cancel: (I) 2a + 6b + c = 5 -2*(II) -2a - 2b + 2c = 2 ---------------------------- 0 + 4b + 3c = 7 Do this kind of thing to eliminate a again. I notice that equation (II) minus equation (III) will eliminate a and give me: 0 - 7b - 3c = -1 ------> 7b + 3c = 1 Now we want to use our two new equations to eliminate either b or c. Since our first equation minus the second one will get rid of c, that's what I'll do. (If for some reason you really wanted to get rid of b at this point, you could multiply the first equation by 7 and the second equation by -4 and add.) Eliminating c we get -3b = 6, so b = -2. Use this in one of the equations with only b and c to find c and then use your b and c in one of the original 3 equations to find a. Technique 2: First take one of the equations and solve for one of the variables: I will take equation (II) and solve for a to get: a = -1 - b + c. Now, take one of the other equations and solve for the same variable: I will take equation (III) and solve for a to get: a = -8b - 2c. Put these two together to get: -1 - b + c = -8b - 2c. Now collect like terms to get: -1 + 7b + 3c = 0 Solve this equation for one variable: b = (1-3c)/7. Replace b by this expression in one of the two equations for a above: i.e., take a = -1 - b + c and replace b to get: a = -1-[(1-3c)/7]+c -----> a = (6+10c)/7 Finally, go back to one of your original equations (I'll use (II)) and replace a and b with the new expressions in c. Take the equation a+b-c = -1 and replace a and b to get: (6+10c)/7 + (1-3c)/7 - c = -1. Now, this is an equation in only c, so you should be able to solve it for c. Once you have c, plug it in to the equations a = (6+10c)/7 and b = (1-3c)/7 to find a and b. I hope this helps you. -Doctor Tracy, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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