Explanation Given Information: The provided system of equation is, x + y + 2 z = − 1 2 x + 2 y + 2 z = 2 0.75 x + 0.75 y + z = 0.25 − x − 2 z = 21 Formula Used: Elementary row operations: Type 1: Replace the row R i by a R i , where a is a nonzero real number. Type 2: Replace the row R i by a R i ± b R j , where a and b is a nonzero real number. Gauss Jordan reduction method: Step 1: First clear the fractions or decimals if any, using operations of type 1. Step 2: Select the first nonzero element of the first row as a pivot. Step 3: Make all elements below and above the pivot element zero in that column zero using operations of type 2. Step 4: Select the first nonzero element in the second row a pivot and clear its column. Step 5: Turn all the selected pivot elements into 1 using operations of type 1. Calculation: Consider the provided system of equations, x + y + 2 z = − 1 2 x + 2 y + 2 z = 2 0.75 x + 0.75 y + z = 0.25 − x − 2 z = 21 The augmented matrix for the given system of equations is, [ 1 1 2 − 1 2 0.75 − 1 2 0.75 0 2 1 − 2 2 0.25 21 ] Apply Gauss Jordan reduction method to get the solution of the given system of equation. Begin by the simplification of the second and third row. Perform the operations R 2 → 1 2 R 2 and R 3 → 4 R 3 , [ 1 1 2 − 1 2 0.75 − 1 2 0.75 0 2 1 − 2 2 0.25 21 ] → [ 1 1 2 − 1 1 3 − 1 1 3 0 1 4 − 2 1 1 21 ] Next pivot the first nonzero element of the first row and clear its column. The pivot element in the first row is 1. Perform the operation R 2 → R 2 − R 1 and R 3 → R 3 − 3 R 1 [ 1 1 2 − 1 1 3 − 1 1 3 0 1 4 − 2 1 1 21 ] → [ 1 1 2 − 1 0 0 − 1 0 0 0 − 1 − 2 − 2 2 4 21 ] Next clear the left over the element of that column. Perform the operation R 4 → R 4 + R 1 . [ 1 1 2 − 1 0 0 − 1 0 0 0 − 1 − 2 − 2 2 4 21 ] → [ 1 1 2 − 1 0 0 0 0 0 1 − 1 − 2 0 2 4 20 ] Next, simplify the third row. Perform the operation R 3 → 1 2 R 3 , [ 1 1 2 − 1 0 0 0 0 0 1 − 1 − 2 0 2 4 20 ] → [ 1 1 2 − 1 0 0 0 0 0 1 − 1 − 1 0 2 2 20 ] Next pivot the first nonzero element of the second row and clears its column

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Author: Stefan Waner, Steven Costenoble

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Chapter 3.2, Problem 31E

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To calculate: The solution of the system of equations x+y+2z=−1, 2x+2y+2z=2, 0.75x+0.75y+z=0.25, −x−2z=21 by the use of Gauss Jordan row reduction.

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Chapter 3.2, Problem 30E

Chapter 3.2, Problem 32E

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The solution of the system of equations x + y + 2 z = − 1 , 2 x + 2 y + 2 z = 2 , 0.75 x + 0.75 y + z = 0.25 , − x − 2 z = 21 by the use of Gauss Jordan row reduction.

Solution Summary: The author calculates the solution of the system of equations by Gauss Jordan row reduction.

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To calculate: The solution of the system of equations x+y+2z=−1, 2x+2y+2z=2, 0.75x+0.75y+z=0.25, −x−2z=21 by the use of Gauss Jordan row reduction.

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