3 Equations, 3 VariablesDate: 01/26/98 at 15:20:34 From: Greg & Kelly Remor Subject: Solving 3 Equations in 3 Variables Dr. Math: I am really rusty with my algebra problems. How do you solve a system of three equations in three variables? 1121 = 25a + 5b + c 626 = 49a + 7b + c 967 = 81a + 9b + c The answers are: a = 104.5 b = -1501.5 c = 6016 Thanks for your help! Date: 01/28/98 at 12:54:18 From: Doctor Jaffee Subject: Re: Solving 3 Equations in 3 Variables Dear Greg and Kelly, This problem looks much like the problems in which you are given three points on a parabola and you have to calculate the equation of the parabola. In this case the points would be (5,1121),(7,626), and (9,967). Since the general equation of a parabola is y = ax^2 + bx + c, if you substitute the x and the y numbers into this equation for each pair of numbers, the result will be exactly the three equations you submitted. But maybe this is more information than you really want to know. Let's get down to actually solving the system. The easiest method I know is the "elimination method". What we want to do is combine two of the equations in such a way that one of the variables is eliminated, then take another pair of the original equations and combine them in a way to eliminate the same variable. I'll show you what I mean. Since there is a "c" at the end of each equation, that would be the easiest variable to eliminate. So, take the first two equations, 1121 = 25a + 5b + c 626 = 49a + 7b + c When you multiply the second equation through by -1 you get -626 = -49a + -7b + -c and when you add that to the first equation you get *495 = -24a + -2b Then work with the 2nd and 3rd equations: 626 = 49a + 7b + c 967 = 81a + 9b + c multiply the first equation through by -1 and get -626 = -49a + -7b + -c then add that to the second equation with the result *341 = 32a + 2b Now the two equations with the asterisks only have two variables and we can eliminate the "b" by adding these two equations: 495 = -24a + -2b 341 = 32a + 2b 836 = 8a Now divide both sides by 8 and the result is a = 104.5. Substitute that back into one of the equations that just has the variables a and b and you should get b = -1501.5. Finally, go back to the original equations, pick any one, and substitute your values for a and b and you will get c = 6016. Now getting back to the problem that I discussed originally, the equation of the parabola would be y = 104.5x^2 + -1501.5x + 6016. I hope I've helped in clearing off some of your rust. Good Luck. -Doctor Jaffee, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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