Algebra I : Semester II : Solving Systems

Sections:

Introduction  |   Section 1  |   Section 2  |   Section 3  |  Section 4  |  Section 5

Section Two

Part 1  |  Part 2  |  Part 3  |  Part 4

Algebra 1: Section 2: Solving Systems Solving with Elimination   Click here for a presentation on the Elimination Method Now let’s put some of the things we found out about the elimination method in a form to remember.  The Elimination Method Check by adding the equations to see if a variable can be eliminated. If not, try subtraction. Change the terms in the second equation to the opposite and check to see if a variable can be eliminated. Solve for the remaining variable. Substitute the value into one of the original equations and solve for the other variable. Write the solution as an ordered pair, (x, y). Example 1:  Using Elimination Use the elimination method to solve the system. 6u + 5v = -2 6u + 9v = 18 Check addition first: 6u + 5v = -2 6u + 9v = 18 12u + 14v = 16 As no variable can be eliminated, subtract by changing all of the terms in the second equation to the opposite.  6u + 5v = -2 -6u – 9v = -18  0u – 4v = -20      -4v = -20 v = -20/-4 = 5 Now substitute v = 5 into the first equation. 6u + 5v = -2 6u + 5(5) = -2 6u + 25 = -2 6u = -2 – 25 6u = -27 u = -27/6 = -4.5 The solution is (-4.5, 5) Check: 6u + 5v = -2 6(-4.5) + 5(5) = -2 -27 +25 = -2 -2 = -2  √ 6u + 9v = 18 6(-4.5) + 9(5) = 18 -27 + 45 = 18 18 = 18  √ Quick Practice Solve each of the following systems using elimination. Check your solution and write it as an ordered pair.  System Solution Details   a + b = 9 a – b = 13       2x + y = 7 -3x + y = -8       -2w – z = -7 2w – z = 7       4m – 2n = 7 4m – 2n = 9       2x – 2y = 8 -2x + 2y = -8       System Solution Details   a + b = 9 a – b = 13 (11, -2) a + b = 9 a – b = 13 2a = 22 a = 11 a + b = 9 11 + b = 9 b = 9 – 11 b = -2 2x + y = 7 -3x + y = -8 (3, 1) 2x + y = 7 3x – y = 8 5x = 15 x = 3 2x + y = 7 2(3) + y = 7 6 + y = 7 y = 7 – 6 = 1 -2w – z = -7 2w – z = 7 (3.5, 0) -2w – z = -7  2w – z = 7       -2z = 0         z = 0/2 = 0 -2w – 0 = -7 -2w = -7 w = -7/-2 = 3.5 4m – 2n = 7 4m – 2n = 9 No solution  4m – 2n = 7 -4m + 2n = -9  0m + 0n = -2 0 = -2 This is a false statement, so there is no solution. 2x – 2y = 8 -2x + 2y = -8 Infinitely many solutions 2x – 2y = 8 -2x + 2y = -8  0x + 0y = 0          0 = 0 This is a true statement, so there are infinite solutions. They are the same equation. Now go on to the next part

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