Systems of Linear Equations Solving by Elimination (Addition)

Homework Answers - Substitution (2,1) (4,3) (1,2) (-4,3) (3,-2) (2,3) (-3,-6) (3,-4) (0,-2) (0,-3) 11. (7,-2) 12. (-4,-1) 13. No Solution 14. (-3,-2) 15. (-3,2) 16. (7,6) 17. (-2,1) 18. (3,2) 19. (-5,1) 20. (4,1)

Racing Review Is (3,4) a solution to the following system of equations? 4x-2y=4 y=4x-8 4(3)-2(4)=4 12-8=4 4=4 OK 4=4(3)-8 4=12-8 YES!

Racing Review What is the solution to this system of equations (use substitution)? x-6y=3 3y=x-3 By substitution… Isolate variable: x=3+3y Plug into other eqn: 3+3y-6y=3 3-3y=3 -3y=0 y=0 Plug ans. into eqn x-6(0)=3 x=3 (3,0)

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NOTES The method is very similar to solving a simple equation. The last way to solve a system of equations that we will consider is by the ELIMINATION or ADDITION method. The method is very similar to solving a simple equation.

Notes - Example 1 Solve the following system of equations: y=12+6x & -y+x=3 1) Line the equations up like an addition problem, making sure the x’s, y’s, and numbers are all together. Then, add the the equations just like you add numbers - remember, you can only combine like terms! The goal is to get rid of one of the variables. y=12+6x  y-6x=12 y - 6x = 12  -y + x = 3 0 - 5x = 15

Notes - Example 1, cont. From previous step: 0 - 5x = 15 2) Solve for the variable that is now alone. -5x=15 (divide both sides by -5) x=-3 3) Plug this value into one of the original equations to find the missing value y=12+6x  y=12+6(-3)  y=-6 ANSWER: (-3,-6)

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Notes - Example 2 Solve the following system of equations: 2x-3y=6 & x+y=-12 Line the equations up, and get ready to add! 2x - 3y = 6  x + y = -12 Now what do you do? How do you add the equations when the numbers don’t easily cancel out? You need to multiply one of the equations by a number so that one of the variables will cancel out when you add. What number do you think we should multiply by, and which equation???

Notes - Example 2, cont. Right! You can multiply the second equation by -2 to get rid of the x’s or 3 to get rid of the y’s. Let’s multiply by -2. Remember, what you do on one side, you must do on the other. 2x - 3y = 6  -2*(x + y) = (-12)*-2  -2x - 2y = 24 0 - 5y = 30 Solve for x (divide both sides by -5)  y=-6 Plug x into one of the equations  x+y=-12  x+(-6)=-12  x=-6 ANSWER: (-6,-6)

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Classwork! :) Find the solution to the following systems of equations by elimination. A B C y=x+1 2y=3x y=5x-1 2y=3x+12 y=3x+1 4y=12x+4 Check your answers!

Classwork! :) Find the solution to the following systems of equations by elimination. A B (2,3) (2,9) C Infinite number of solutions - they are the same line! Check your answers!

Homework Answers - Elimination (6,-6) (7,-1) (10,-1) (-1,-1) (-1,3) (-1,-8) (5,6) (6,-9) (1,4) (9,5) (-4,-4) (2,-5) 13. (0,-1) 14. (4,-2) 15. (8,-1) 16. (0,-2) 17. (1,-2) 18. (-6,0) 19. (-1,-5) (-2,-4) (-6,4) Infinite # of Solutions b/c lines are the same (2,0) (-1,1)