SolvingSystems.Elimination

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The elimination method involves (surprise) eliminating a variable. I personally find this method the best but it is situational as the 2 equations have to have identical or inverse terms. Inverse terms are the best, you add them. Identical or similar terms you subtract. ** ﻿ ** // Manipulate the equations to derive inverse or identical terms for one of the variables and arrange the terms for elimination. //  -8x+3y=12 8x-9y=12 **You can add inverse terms.** //Since 8x and -8x are inverse terms,// ﻿// adding the equations together will // ﻿ // elimiate them**.** // We add the two equations together and are left with -6y=24 After that you solve like a regular equation; divide each side by -6 and you get y=-4. Then all you have to do is substitute y=-4 into one of the original equations. 8x-9(-4)=12. The answer is x=-3. Like all methods, we then check. 8(-3)-9(-4)=12 -24+36 //is// 24, so (-3,-4) is the correct answer.
 * The Elimination Method. **
 * Solving using the Elimination method with addition. **

2x+4y-3=15 2x+2y-1=13 // You then take the bottom equation and subtract it from the top. // You get 2y-2=2. Add 2 to each side and we are left with 2y=4, so y=2. // Going back to the top, we plug in the value for y to find x. // 2x+4(2)-3=15 2x=10 We find that x=5, so we check using the other equation in the system. 2(5)+2(2)-1=13. 10+4-1=13  13=13.  Therefore, (5,2) is our answer.
 * Subtracting in Elimination is not much different. **

// If you don't have identical terms, you can manipulate the system to produce inverse or identical terms. // 5x+4y=7 3x+2y=5 In this case we will multiply the second equation by -2 to get inverse terms. -6x-4y=-10 // We then add this to the first equation, just like when we add them regularly. // x=3 is what we .are left with You then plug in -3 to where x is supposed to be in one of the equations above 3(3)+2y=5 9+2y=5 2y=-4 y=-2. We find that y=-2, so we use both of our variables in the other equation to check. 5(3)+4(-2)=7 15-8=7  7=7  (3,-2) is our solution. 3x+5y=21 -2x+2y=2 Multiply the first one by 2 and the second one by 3. 6x+10y=42 -6x+6y=6 Now add to eliminate. 16y=48 Divide each sid eby 16 and we get y=3. Now plug in the value for y into one of the equations to find x. -2x+2(3)=2, we get x=2. Use the other equation to check 3(2)+5(3)=21 is correct, so our answer is (2,3)
 * Advanced Elimination. **
 * You can also multiply both equations if needed. **

All that this requires you to do is subtract. x=-5y+41 x=-3y+23 We subtract these and we get 0=-2y+18. From this we see y=9, and we then plug it in to the equations above for x x=-3(9)+23 Then we see that x=4. After that, check. -4=-5(9)+41 is correct, so the answer is (-4,9)
 * Using Elimination with Slope Intercept form **

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