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2-2 - Chapter 2.2 Gauss-Jordan Elimination Unique Solutions...

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Chapter 2.2: Gauss-Jordan Elimination Unique Solutions While the method of substitution works, it is somewhat clumsy when we work with equations involving three or more variables. We will look at a more methodical way of solving a system of equations, called the Gauss-Jordan Elimination Method . Two systems of equations are equivalent if they have the same solution. With the Gauss-Jordan method, we transform a given system into an simpler equivalent system using three operations: 1. Interchanging equations. 2. Multiplying every term of an equation by a constant. 3. Adding/subtracting two equations. Example: Solve the following system: x + y = 16 8 x + 10 y = 146 Subtract 8 times equation (1) from equation (2): x + y = 16 2 y = 18 Multiply equation (2) by 1 2 : x + y = 16 y = 9 Subtract equation (2) from equation (1): x = 7 y = 9 We have solved for x and y , and found one solution to the given system of equations. (Don’t forget to check the solution!) What did I do to find my solution? 1. Used an x in the first equation to get rid of the x in the second equation. 1
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2. Used the leftover y in the second equation to get rid of the y in the first equation. This same process will work for more than two variables. We will first get rid of the x , then get rid of y , then get rid of z . Example: Find the solution to the given system: x + y + z = 15 x + 3 y + 4 z = 9 - x + 2 y - z = 3 Subtract equation 1 from equation 2: x + y + z = 15 + 2 y + 3 z = - 6 - x + 2 y - z = 3 Add equation 1 to equation 3: x + y + z = 15 2 y + 3 z = - 6 3 y = 18 Interchange equation 2 and equation 3: x + y + z = 15 3 y = 18 2 y + 3 z = - 6 Multiply equation 2 by 1 3 : x + y + z = 15 y = 6 2 y + 3 z = - 6 Subtract equation 2 from equation 1: x + z = 9 y = 6 2 y + 3 z = - 6 Subtract two times equation 2 from equation 3: 2
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x + z = 9 y = 6 3 z = - 18 Multiply equation 3 by 1 3 : x + z = 9 y = 6 z = - 6 Subtract equation 3 from equation 1: x = 15 y = 6 z = - 6 We have found that x = 15, y = 6, z = - 6 is the solution of the given system. Remember to check your answers! We used the x in the first equation to get rid of x in the other equations. We used the y in the second equation to get rid of y in the other equations. We used the z in the third equation to get rid of z in the other equations. The variables really were just placeholders – we had all x , y , and z terms lining up vertically. We can use a matrix to reduce the amount of writing we have to do. A matrix is just an array of numbers. Each row will represent an equation, and each column will represent an x term, y term, z term, or constant term.
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