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Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

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BuyFindarrow_forward

Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

In Exercises 1-42, use Gauss-Jordan row reduction to solve the given systems of equation. We suggest doing some by hand and others using technology. [HINT: See Examples 1-6.]

x + y + 4 w = 1

2 x 2 y 3 z + 3 w = 1

4 y + 6 z + w = 4

2 x + 4 y + 9 z = 6

To determine

To calculate: The solution of the given system of equations x+y+4w=1,2x2y3z+3w=1,4y+6z+w=4,2x+4y+9z=6 by the use of Gauss Jordan row reduction.

Explanation

Given Information:

The system of equation is,

x+y+4w=12x2y3z+3w=14y+6z+w=42x+4y+9z=6

Formula Used:

Elementary row operations

Type 1: Replacing the row Ri by aRi, where a is a nonzero number.

Type 2: Replacing the row Ri by aRi±bRj, where a is a nonzero 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 pivot.

Step 3: Use the pivot to clear its column using operations of type 2.

Step 4: Select the first nonzero element in the second row a pivot and clear its column.

Step 5: Select the first nonzero element in the third row a pivot and clear its column.

Step 6: Select the first nonzero element in the fourth row a pivot and clear its column.

Step 7: Turn all the selected pivot elements into a 1 using operations of type 1.

Calculation:

Consider the system of equation,

x+y+4w=12x2y3z+3w=14y+6z+w=42x+4y+9z=6

The augmented matrix for the given system of equations is,

[11041202244369310146]

Apply Gauss Jordan reduction method to get the solution of the given system of equation.

Begin by, the selection of the pivot as the first nonzero element of the first row and clear its column.

Perform the operation R2R22R1 and R4R42R1

[11041202244369310146][11041000442369618344]

Next pivot the first nonzero element of the second row and clear its column.

Perform the operation R14R1+R2 and R3R3+R2, [11041000442369618344][403101000402339658314]

Next clear the leftover element of that column..

Perform the operation R42R4+R2

[403101000402339658314][40310100040033156522315]

Next pivot the first nonzero element of the third row and clear its column.

Perform the operation R1R1+R3 and R2R2+R3, [40310100040033156522315][40052000400031511522215]

Next clear the leftover element of that column.

Perform the operation R4R45R3

[40052000400031511522215][400520004000301153210]

Next simplify the fourth row

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