Question

Test the consistency of the following system of linear equations: 2x + 3y + 5z + t = 3 3x + 4y + 2z + 3t = −2 x + 2y + 8z − t = 8 7x + 9y + z + 8t = 0

Answer 1

Explanation:

Given:

$2x + 3y + 5z + t = 3 \\ 3x + 4y + 2z + 3t = −2 \\ x + 2y + 8z − t = 8 \\ 7x + 9y + z + 8t = 0$

To find: Test the consistency of the following system of linear equations.

Solution:

To test the consistency of the system of linear equations write the system in matrix form.

AX=B

and then calculate |A|,if determinant of A is zero than system of linear equations have no solution.i.e. not consistent.

$\left[\begin{array}{cccc}2&3&5&1\\3&4&2&3\\1&2&8&-1\\7&9&1&8\end{array}\right]\left[\begin{array}{cccc}x\\y\\z\\t\end{array}\right]=\left[\begin{array}{cccc}3\\-2\\8\\0\\\end{array}\right]$

Find the determinant of A

$|A|=\left|\begin{array}{cccc}2&3&5&1\\3&4&2&3\\1&2&8&-1\\7&9&1&8\end{array}\right|$

Expand the determinant along R1

$|A|=2\left|\begin{array}{ccc}4&2&3\\2&8&-1\\9&1&8\end{array}\right|-3\left|\begin{array}{ccc}3&2&3\\1&8&-1\\7&1&8\end{array}\right|+5\left|\begin{array}{ccc}3&4&3\\1&2&-1\\7&9&8\end{array}\right|-1\left|\begin{array}{ccc}3&4&2\\1&2&8\\7&9&1\end{array}\right|$

Solve the determinant, we will get that

|A|=0

Note*: One can apply properties of determinants to simplify the calculation.

Thus,

The system of linear equations is inconsistent and have no unique solution.

Hope it helps you.

To learn more on brainly:

solve 2x + y + 6z = 9; 8x + 3y + 2z = 13; x + 5y + z = 17 by using gauss Seidel iteration method

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