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Solving Systems of Equations using Elimination - Concept
A system of equations is two or more equations that contain the same variables. Solving systems of equations by elimination is one method to find the point that is a solution to both (or all) original equations. Besides solving systems of equations by elimination, other methods of finding the solution to systems of equations include graphing, substitution and matrices.
By this point in your Maths career, you guys are all really good at solving one equation with one variable. A system of equations however, has two equations and two variables. So you want to somehow condense them so instead of having two equations you only have one equation. One way you can do that is by using what's called elimination.
If I have two equations with two variables, I'm going to look for coefficients that are additive inverses. Remember additive inverses mean the two numbers add up to 0 like 2 and -2 or 1.5 and -1.5. They have to have a positive sign and a negative sign but still be the exact same number. So if I have two coefficients that are additive inverses I can add the equations vertically and one of my variables will be eliminated.
I'm going to show you one problem that's going to start it off for you to show you what I mean. I'm just making this up 3x+4y=10 I just made that up and 2x-4y=14, I don't know. Let's just say this was a system of equations and I was asked to solve using elimination. If I look the coefficients here for y +4 and -4 are called additive and versus they add up to 0. If I add these equations vertically 5x I'll have 0y, 5x=24. Now this is one equation with one variable that I can solve really easily and go through to find solution to my system of equations.
So you guys when you're given a problem and asked to use elimination, you're wanting to looking for additive inverses. Sometimes you have to do some clever multiplying, let me show what I mean. Let's just say that instead of +4 and -4 I had something else. Let's see 3x+4y=10. Let's say I had 3x, no I'm just making this up so you got to help me out a second guys. Let's say I had 2x-2y=5 or something, I don't have any additive inverses nothing that I can add together to make a variable cancel out. However, I could multiply the second equation by 2 and here is why that's a good idea. If I multiply all of these coefficients by 2, then that would become a 4, that would become a -4, that would become a 10 and now my coefficients of y are additive inverses. That's an okay Mathematical process I'm allowed to multiply by 2 as long as I multiply every term in the equation by 2.
So I'm going to leave you guys with the idea that to use elimination you're looking for coefficients that are additive inverses and then using those you can add your equations vertically to have one letter and one equation.