What is the solution of the system of equations?
3x + 2y + z = 7, 5x + 5y + 4z = 3, and 3x + 2y + 3z = 1

Solution:

Given system of equations:

3x + 2y + z = 7, 5x + 5y + 4z = 3, and 3x + 2y + 3z = 1

A system of linear equations is a set of equations which are satisfied by the same set of variables.

Write an augmented matrix using the variable coefficients and constants separately

Perform row operations until you obtain an identity matrix.

|5 5 4||3|

|3 2 3||1|

|3 2 1||7| …… (1)

eq(1) is the augmented matrix

Subtract row 2 from row 1-

We get as follows:

|2 3 1||2|

|3 2 3||1|

|3 2 1||7|

Subtract row 3 from row 1- we get as follows:

|-1 1 0||-5|

|3 2 3||1|

|3 2 1||7|

Multiply row 1 by 3 and add to row 2 - we get as follows:

|-1 1 0||-5|

|0 5 3||-14|

|3 2 1||7|

Multiply row 1 by 3 and add to row 3 - we get as follows:

|-1 1 0||-5|

|0 5 3||-14|

|0 5 1||-8|

Subtract row 2 from row 3 - we get as follows:

|-1 1 0||-5|

|0 5 3||-14|

|0 0 -2||6|

Divide row 3 by -2 - we get as follows:

|-1 1 0||-5|

|0 5 3||-14|

|0 0 1||-3|

Multiply row 3 by -3 and add to row 2 - we get as follows:

|-1 1 0||-5|

|0 5 0||-5|

|0 0 1||-3|

Divide row 2 by 5 - we get as follows:

|-1 1 0||-5|

|0 1 0||-1|

|0 0 1||-3|

Multiply row 1 by -1 - we get as follows:

|1 -1 0||5|

|0 1 0||-1|

|0 0 1||-3|

Add row 2 to row 1 - we get as follows:

|1 0 0||4|

|0 1 0||-1|

|0 0 1||-3|

Rank of augmented matrix is the same as the rank of the constant matrix, which is also equal to the number of unknowns.

There is a unique solution.

The solution is x = 4, y = -1 and z = -3.

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What is the solution of the system of equations? 3x + 2y + z = 7, 5x + 5y + 4z = 3, and 3x + 2y + 3z = 1

Summary:

The solution of the system of equations 3x + 2y + z = 7, 5x + 5y + 4z = 3, and 3x + 2y + 3z = 1 is x = 4, y = -1 and z = -3.