Question

Solve the given system of equation by Gauss Elimination method. 3x + 4y – z = -6 -2y + 10z = -8 4y – 2z = -2

Answer 1

Step-by-step explanation:

Given:

System of linear equations

$3x + 4y - z = - 6 \\ - 2y + 10z = - 8 \\ 4y - 2z = - 2$

To find: Solve the given system of equation by Gauss Elimination method.

Solution:

In Gauss Elimination method the coefficient matrix should be converted into Row-echelon form.

Step 1: Write the equations in form of matrix

$\left[\begin{array}{ccc}3&4&-1\\0&-2&10\\0&4&-2\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] =\left[\begin{array}{ccc}-6\\-8\\-2\end{array}\right]$

Step 2: Apply elementary row operations on matrices.

$R_3->R_3+2R_2$

$\left[\begin{array}{ccc}3&4&-1\\0&-2&10\\0&0&18\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] =\left[\begin{array}{ccc}-6\\-8\\-18\end{array}\right]$

Step 3: Find value of z.

It is clear from third row

$18z = - 18$

$\bf z = - 1$

Step 4: Find value of y.

From second row

$- 2y + 10z = - 8$

put value of z

$- 2y - 10 = - 8 \\ - 2y = - 8 + 10$

$- 2y = 2$

$\bf y = - 1$

Step 5: Find the value of x.

Write eq from row 1

$3x + 4y - z = - 6$

put values of y and z

$3x + 4( - 1) - ( - 1) = - 6$

$3x = - 6 + 3$

$3x = - 3$

$\bf x = - 1$

Final answer:

$x = - 1 \\ y = - 1 \\ z = - 1$

Hope it will help you.

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