Monday, August 21, 2017

College Algebra, Chapter 7, 7.1, Section 7.1, Problem 42

Find the complete solution of the system

{4xy+36z=24x2y+9z=32x+y+6z=6


We first write the augmented matrix of the system and using Gauss-Jordan Elimination.

[41362412932166]

14R1

[1149612932166]

R3+2R1R3

[114961293032126]

23R3

[1149612930184]

R2R1R2

[114960941890184]

49R2

[1149601840184]

R3R2R3

[1149601840000]

R114R2R1

[107501840000]


This is in reduced row-echelon form since the last row represents the equation 0=0, we may discard it. So the last matrix corresponds to the system


{x7z=5y8z=4


To obtain the complete solution, we solve for the leading variables x and y in terms of the nonleading variable z and we let z be any real numbers. Thus, the complete solution is


x=7t5y=8t4z=t


where t is any real number.

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