Question

Solve x +2y +3z =4, 2x +3y +8z =7 and x –y -9z =1 using Gauss-Jordan method

Answer 1

Step-by-step explanation:

X+ 2y +3z=4

2x + 3y +8z = 7

x - y - 9z = 1

Answer 2

The solution of equations are

$\bf x = 2 \\ \bf y = 1 \\\bf z = 0$

Given:

• A system of linear equations.
• $x +2y +3z =4, \\ 2x +3y +8z =7 \\ x –y -9z =1$

To find:

• Solution of equations using Gauss-Jordan Elimination.

Solution:

Concept to be used:

Gauss-Jordan Elimination: Form matrix with coefficients of x,y and z and constant term.

Step 1:

Let

$A=\left[\begin{array}{ccc}1&2&3\\2&3&8\\1&-1&-9\end{array}\right]$

and

$B=\left[\begin{array}{ccc}4\\7\\1\end{array}\right]$

Step 2:

Form augmented matrix.

C=[A|B]

$C=\left[\begin{array}{ccc}1&2&3& 4\\2&3&8& 7\\1&-1&-9&1\end{array}\right]$

Using elementary row operations convert A in identity matrix,thus result will be obtained in B.

$R_2=R_2-2R_1$

$C=\left[\begin{array}{ccc}1&2&3& 4\\0& - 1&2& - 1\\1&-1&-9&1\end{array}\right]$

$R_3=R_3-R_1$

$C=\left[\begin{array}{ccc}1&2&3& 4\\0& - 1&2& - 1\\0&-3&-12& - 3\end{array}\right]$

$R_2=-R_2$

$C=\left[\begin{array}{ccc}1&2&3& 4\\0& 1& - 2& 1\\0&-3&-12& - 3\end{array}\right]$

$R_1=R_1-2R_2$

$C=\left[\begin{array}{ccc}1&0&7& 2\\0& 1& - 2& 1\\0&-3&-12& - 3\end{array}\right]$

$R_3=R_3+3R_2$

$C=\left[\begin{array}{ccc}1&0&7& 2\\0& 1& - 2& 1\\0&0&-18& 0\end{array}\right]$

$R_3=-\frac{1}{18}R_3$

$C=\left[\begin{array}{ccc}1&0&7& 2\\0& 1& - 2& 1\\0&0&1& 0\end{array}\right]$

$R_2=R_2+2R_3$

$C=\left[\begin{array}{ccc}1&0&7& 2\\0& 1& 0& 1\\0&0&1& 0\end{array}\right]$

$R_1=R_1+7R_3$

$C=\left[\begin{array}{ccc}1&0&0& 2\\0& 1& 0& 1\\0&0&1& 0\end{array}\right]$

Thus,

$\bf x = 2 \\ \bf y = 1 \\ \bf z = 0$

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Learn more:

1) i. by using Cramer's rule and Matrix inversion method, when the coefficient matrix is nonsingular

ii. by using Gauss-Jordan method. Also determine whether the system has a unique solution or infinite number of solutions or no solution. Find the solution if exist.

5x - 6y + 4z = 15, 7x + 4y - 3z = 19,2x + y + 6z = 46

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

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