10/21/13 Solving Systems by Elimination 35 EXAMPLE: 5x + 2y = 1 x 3y = 7 1.Multiply the Top equation by the coefficient of the x on the bottom equation and write that equation next to the first equation 2. Next, multiply the bottom equation by the negative of the coefficient of the top x. and write that equation next to the bottom equation 3. Add the new equations to Eliminate the x s. 4. Divide by the coefficient of y to get y. 5. Substitute y= -2 into one of the equations to solve for x. 6. The answer is the ordered pair (1, -2). 1. 1(5x + 2y = 1) 5x + 2y = 1 5x + 2y = 1 5x + 2y = 1 x 3y = 7 2.-5(x 3y = 7) -5x + 15y =-35 5x + 2y = 1 5x + 2y = 1 x 3y = 7-5x+ 15y =-35 3. 5x + 2y = 1-5x+ 15y =-35 17y = -34 4. 17y = -34 y = -2 17 17 5. x 3(-2) = 7 x + 6 = 7-6 -6 x = 1
Systems of Equations in Two Variables
A system of Equations has at least two equations and two variables. A system could have thousands of equations and variables, but we will deal with only two for now.
A system in two variables will have an ordered pair for the solution, (x,y). There are several different ways to find the answer, in this video we will learn the method called Elimination.
Solving a System by Elimination Given two Equations, you will: 1. Eliminate one of the variables 2. Solve for the remaining variable 3. Substitute the one you now know into one of the equations and solve for the other. Always eliminate the X first, find the Y and then Plug in the Y to find the X.
Eliminating the X First Given the System: x + 3y = 2 4x + 5y = 1
x + 3y = 2 4x + 5y = 1 1.Multiply the Top equation by the coefficient of the x on the bottom equation. x + 3y = 2 4x + 5y = 1 4(x + 3y = 2) 4x + 12y =8
x + 3y = 2 4x + 5y = 1 4x + 12y =8 1. Multiply the Top equation by the coefficient of the x on the bottom equation. 4(x + 3y = 2) 4x + 12y =8 2. Rewrite that equation next to the first equation 4x + 12y =8
x + 3y = 2 4x + 12y =8 4x + 5y = 1 1. Multiply the Top equation by the coefficient of the x on the bottom equation. 4(x + 3y = 2) 4x + 12y =8 2. Rewrite that next to the first equation 4x + 12y =8 3. Next, multiply the bottom equation by the negative of the coefficient of the top x. -1(4x + 5y = 2) -4x -5y = -2
x + 3y = 2 4x + 12y =8 4x + 5y = 1-4x - 5y =-1 1. Multiply the Top equation by the coefficient of the x on the bottom equation. 4(x + 3y = 2) 4x + 12y =8 2. Rewrite that next to the first equation 4x + 12y =8 3. Next, multiply the bottom equation by the negative of the coefficient of the top x. -1(4x + 5y = 1) -4x -5y = -1 4. Write that equation next to the bottom equation. -4x - 5y =-1
x + 3y = 2 4x + 12y =8 4x + 5y = 1-4x - 5y =-1 1. Multiply the Top equation by the coefficient of the x on the bottom equation. 4(x + 3y = 2) 4x + 12y =8 2. Rewrite that next to the first equation 4x + 12y =8 3. Next, multiply the bottom equation by the negative of the coefficient of the top x. -(4x + 5y = 2) -4x -5y = -1 4. Write that equation next to the bottom equation -4x - 5y =-1 5. Add the new equations to Eliminate the x s. 4x + 12y =8-4x - 5y =-1 7y = 7
x + 3y = 2 4x + 12y =8 4x + 5y = 1-4x - 5y =-1 1. Multiply the Top equation by the coefficient of the x on the bottom equation. 4(x + 3y = 2) 4x + 12y =8 2. Rewrite that next to the first equation 4x + 12y =8 3. Next, multiply the bottom equation by the negative of the coefficient of the top x. -(4x + 5y = 2) -4x -5y = -1 4. Write that equation next to the bottom equation -4x - 3y =-1 5.Add the new equations to Eliminate the x s. 4x + 12y =8-4x - 5y =-1 7y = 7 6. Divide by the coefficient of y to get y. 7y = 7 7 7 y=1
x + 3y = 2 4x + 5y = 1 Now plug in for y (y=1) and solve for x x + 3y = 2 y = 1 x + 3(1) = 2 x + 3 = 2-3 -3 x = -1 The solution is the ordered pair (-1,1).
4x - y = 9 x - 3y = 16 1. Multiply the Top equation by the coefficient of the x on the bottom equation. 1(4x - y = 9) 4x - y = 9 2. Rewrite that next to the first equation 4x - y = 9 4x - y = 9 3. Next, multiply the bottom equation by the negative of the coefficient of the top x. -4(x - 3y = 16) -4x +12y = -64 4. Write that equation next to the bottom equation x - 3y = 16-4x +12y = -64 5.Add the new equations to Eliminate the x s. 4x - y = 9-4x + 12y = -64 11y = -55 6. Divide by the coefficient of y to get y. 11y = -55 11 11 y=-5
4x - y = 9 x - 3y = 16 Now plug in for y (y=-5) and solve for x 4x - y = 9 y = -5 4x - (-5) = 9 4x + 5 = 9-5 -5 4x = 4 x = 1 The solution is the ordered pair (1,-5).
3x 7y = 1 2x - 3y = -1 1. Multiply the Top equation by the coefficient of the x on the bottom equation. 2(3x 7y =1) 6x 14y = 2 2. Rewrite that next to the first equation 3x 7y = 1 6x 14y = 2 3. Next, multiply the bottom equation by the negative of the coefficient of the top x. -3(2x 3y = -1) -6x + 9y = 3 4. Write that equation next to the bottom equation 2x - 3y = -1-6x + 9y = 3 5.Add the new equations to Eliminate the x s. 6x 14y = 2-6x + 9y = 3-5y = 5 6. Divide by the coefficient of y to get y. -5y = 5-5 -5 y=-1
3x 7y = 1 2x - 3y = -1 Now plug in for y (y=-1) and solve for x 2x - 3y = -1 y = -1 2x - 3(-1) = -1 2x + 3 = -1-3 -3 2x = -4 x = -2 The solution is the ordered pair (-2,1).
2x 5y = 13 5x + 3y = 17 10x 25y = 65-10x - 6y =-34 1. Multiply the Top equation by the coefficient of the x on the bottom equation. 5(2x 5y = 13) 10x 25y = 65 2. Rewrite that next to the first equation 2x 5y = 13 10x 25y = 65 3. Next, multiply the bottom equation by the negative of the coefficient of the top x. -2(5x + 3y = 17) -10x - 6y = -34 4. Write that equation next to the bottom equation 5x + 3y = 17-10x -6y = -34 5.Add the new equations to Eliminate the x s. 10x 25y = 65-10x - 6y =-34-31y =31 6. Divide by the coefficient of y to get y. -31y = 31-31 -31 y=-1
2x 5y = 13 5x + 3y = 17 Now plug in for y (y=-1) and solve for x 2x - 5y = 13 y = -1 2x - 5(-1) = 13 2x + 5 = 13-5 -5 2x = 8 x = 4 The solution is the ordered pair (4,-1).