Solve by using the ELIMINATION method The goal is to eliminate one of the variables by performing multiplication on the equations. Multiplication is not always needed. Add the two equations together. Solve the two step equation with one variable. Plug in your answer in one of the original equations to solve for the second variable. Solve a System of Equations

How to eliminate a variable? Example of a system is below: 3x + 4y = 5 2x – 6y = 12 To eliminate the x variable, we can multiply the first equation by -2, and the second by 3. Please note that we could eliminate the y variable if we choose to.

Solve a System of Equations The result would be: -2 [3x + 4y = -8]  -6x – 8y = 16 3 [2x – 6y = 12]  (+) 6x – 18y = y = y = -2 This is only half the answer!

Our partial answer is y = -2 Complete the problem; plug in your answer. The original problem is below 3x + 4y = 5  3x +4 (-2) = 5 2x – 6y = 12 3x - 8 = x = 13 x = 4 1/3 Final solution is (4 1/3, -2)

Solve a System of Equations Now we will completely solve a slightly more easy problem since we are just starting this topic. Sample problem is below: x + y = 7 (first equation) 3x + y = 3 (second equation)

Solve a System of Equations x + y = 7 (first equation) 3x + y = 3 (second equation) The goal is to “eliminate” one of the variables first to solve for the other variable. Variable must have the same number but opposite signs.

Solve a System of Equations x + y = 7 3x + y = 3  multiply bottom equation by (-1) (same number, but signs are not opposite. Fix it!) -1[3x + y = 3]  multiply every term by -1. The new equation is below: -3x – y = -3  modified equation to use. 3x + y = 3 is the same as −3x – y = −3

Solve a System of Equations Now we can add the two equations together. x + y = 7 (first equation that was not touched) (+) -3x – y = -3 (new equation, from multiplying) -2x = 4 (divide by -2 on both sides) x = -2 (this is only half our answer… )

Solve a System of Equations Now, use the original equations to find the second part of your answer. x + y = 7 (first equation) 3x + y = 3 (second equation) (since “x = −2” is part of our answer, use it!) −2 + y = 7 y = 9

Solve a System of Equations For the system of equations problem: x + y = 7 (1 st original equation) 3x + y = 3 (2 nd original equation) Final solution is (-2, 9) This means that these two linear equations intersect at (-2, 9).

Solve a System of Equations To eliminate the x variable, what should you multiply by for the below problem? 2x + 4y = 12 x + 3y = 5 Multiply the bottom equation by -2, and then add the two equations together. You don’t have to touch the top equation at all.

Solve a System of Equations 2x + 4y = 12  2x + 4y = 12 (Did not have to change the top one) -2[x + 3y = 5]  -2x – 6y = y = y = −1 Second step : (plug in your answer in order to find the second variable.) 2x + 4( −1) = 12 2x – 4 = ___________ 2x = 16  x = 8 ; solution is (8, -1) You should practice at least 25 problems. Let’s go to work!!