Linear Algebra and Its Applications, 4th Edition

Published by Brooks Cole
ISBN 10: 0030105676
ISBN 13: 978-0-03010-567-8

Chapter 1 - Section 1.3 - An Example of Gaussian Elimination - Problem Set: 2

Answer

(2, -1) If the right-hand side changes to (4, 44), the new solution is (8, -4)

Work Step by Step

Take the triangular system of equations we found in problem 1: 2x+3y=1 -6y=6 Solve using back-substitution. We can solve equation 2 using basic algebra; -6y=6 6y=-6 y=-1 Now that we know the value of y, we can substitute y=-1 into equation 1 and solve for x 2x+3(-1)=1 2x-3=1 2x=4 x=2 Therefore, the solution to the equation is (2, -1) However, the problem also asks for the new solution if the right-hand side of the equation changes to (4, 44) (The textbook is referring back to the original system, not the triangular system we created in problem one) This creates a new system: 2x+3y=4 10x+9y=44 We can solve this new system using Gaussian Elimination. Multiply equation one by 5, and subtract equation one from equation 2 {10x+9y=44}-{10x+15y=20}={0x-6y=24} Our new system of equations is: 2x+3y=4 -6y=24 Now we can solve using back-substitution: 6y=-24 y=-4 2x+3(-4)=4 2x-12=4 2x=16 x=8 The solution to the new system is (8, -4)
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