# 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 − z = − 2 x − y − 7 z = 0 0.75 x − 0.5 y + 0.25 z = 14 x + y + z = 4

### Finite Mathematics

7th Edition
Stefan Waner + 1 other
Publisher: Cengage Learning
ISBN: 9781337280426

Chapter
Section

### Finite Mathematics

7th Edition
Stefan Waner + 1 other
Publisher: Cengage Learning
ISBN: 9781337280426
Chapter 3.2, Problem 32E
Textbook Problem
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## 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 − z = − 2 x − y − 7 z = 0 0.75 x − 0.5 y + 0.25 z = 14 x + y + z = 4

To determine

To calculate: The solution of the given system of equations,

x+yz=2xy7z=00.75x0.5y+0.25z=14x+y+z=4

By the use of Gauss Jordan row reduction.

### Explanation of Solution

Given Information:

The system of equation is,

x+yz=2xy7z=00.75x0.5y+0.25z=14x+y+z=4

Formula Used:

Elementary row operations

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

Type 2: Replace 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: Turn all the selected pivot elements into a 1 using operations of type 1.

Calculation:

Consider the system of equation,

x+yz=2xy7z=00.75x0.5y+0.25z=14x+y+z=4

The augmented matrix for the given system of equations is,

[111210.75110.5170.2510144]

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

Begin by, the simplification of the third row.

Perform the operation R34R3,

[111210.75110.5170.2510144][11121311217110564]

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

Perform the operation R2R2R1 and R3R33R1

[11121311217110564][11120012516412624]

Next clear the left over element of that column.

Perform the operation R4R4R1.

[11120012516412624][11120002506422626]

Next simplify the second and fourth row.

Perform the operation R212R2 and R412R4, [11120002506422626][11120001503411623]

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

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