Question

Solve 10x - 7y + 3z + 5u = 6, -6x + 8y - 2 - 4u = 5,
3x + y + 4z + llu= 2, 5x - 9y – 2z + 4u = 7 by Gauss Elimination Method:​

Answer 1

Given:

10x - 7y + 3z + 5u = 6

-6x + 8y - 2 - 4u = 5

3x + y + 4z + 11u = 2

5x - 9y - 2z + 4u = 7

To find:

Solve the given equations by Gauss Elimination Method.

Solution:

5u + 10x - 7y + 3z = 6  

-4u - 6x + 8y + 0z = 7  

11u + 3x + y + 4z = 2  

4u + 5x - 9y - 2z = 7

$\left[\begin{array}{cccccc}5&10&-7&3&|&6\\-4&-6&8&0&|&7\\11&3&1&4&|&2\\4&5&-9&-2&|&7\end{array}\right]$

R₁ → R₁ ÷ 5

$\left[\begin{array}{cccccc}1&2&-1.4&0.6&|&1.2\\-4&-6&8&0&|&7\\11&3&1&4&|&2\\4&5&-9&-2&|&7\end{array}\right]$

R₂ → R₂ + 4×R₁

$\left[\begin{array}{cccccc}1&2&-1.4&0.6&|&1.2\\0&2&2.4&2.4&|&11.8\\11&3&1&4&|&2\\4&5&-9&-2&|&7\end{array}\right]$

R₃ → R₃ - 11×R₁

$\left[\begin{array}{cccccc}1&2&-1.4&0.6&|&1.2\\0&2&2.4&2.4&|&11.8\\0&-19&16.4&-2.6&|&-11.2\\4&5&-9&-2&|&7\end{array}\right]$

R₄ → R₄ - 4×R₁

$\left[\begin{array}{cccccc}1&2&-1.4&0.6&|&1.2\\0&2&2.4&2.4&|&11.8\\0&-19&16.4&-2.6&|&-11.2\\0&-3&-3.4&-4.4&|&2.2\end{array}\right]$

R₂ → R₂ ÷ 2

$\left[\begin{array}{cccccc}1&2&-1.4&0.6&|&1.2\\0&1&1.2&1.2&|&5.9\\0&-19&16.4&-2.6&|&-11.2\\0&-3&-3.4&-4.4&|&2.2\end{array}\right]$

R₁ → R₁ - 2×R₂

$\left[\begin{array}{cccccc}1&0&-3.8&-1.8&|&-10.6\\0&1&1.2&1.2&|&5.9\\0&-19&16.4&-2.6&|&-11.2\\0&-3&-3.4&-4.4&|&2.2\end{array}\right]$

R₃ → R₃ + 19×R₂

$\left[\begin{array}{cccccc}1&0&-3.8&-1.8&|&-10.6\\0&1&1.2&1.2&|&5.9\\0&0&39.2&20.2&|&100.9\\0&-3&-3.4&-4.4&|&2.2\end{array}\right]$

R₄ → R₄ + 3×R₂

$\left[\begin{array}{cccccc}1&0&-3.8&-1.8&|&-10.6\\0&1&1.2&1.2&|&5.9\\0&0&39.2&20.2&|&100.9\\0&0&0.2&-0.8&|&19.9\end{array}\right]$

R₃ → R₃ × 0.0255

$\left[\begin{array}{cccccc}1&0&-3.8&-1.8&|&-10.6\\0&1&1.2&1.2&|&5.9\\0&0&1&0.5153&|&2.574\\0&0&0.2&-0.8&|&19.9\end{array}\right]$

R₁ → R₁ + 3.8×R₃

$\left[\begin{array}{cccccc}1&0&0&0.1582&|&-0.8189\\0&1&1.2&1.2&|&5.9\\0&0&1&0.5153&|&2.574\\0&0&0.2&-0.8&|&19.9\end{array}\right]$

R₂ → R₂ - 1.2×R₃

$\left[\begin{array}{cccccc}1&0&0&0.1582&|&-0.8189\\0&1&0&0.5816&|&2.8112\\0&0&1&0.5153&|&2.574\\0&0&0.2&-0.8&|&19.9\end{array}\right]$

R₄ → R₄ - 0.2×R₃

$\left[\begin{array}{cccccc}1&0&0&0.1582&|&-0.8189\\0&1&0&0.5816&|&2.8112\\0&0&1&0.5153&|&2.574\\0&0&0&-0.9031&|&19.3852\end{array}\right]$

R₄ → R₄ ×(-1.1073)

$\left[\begin{array}{cccccc}1&0&0&0.1582&|&-0.8189\\0&1&0&0.5816&|&2.8112\\0&0&1&0.5153&|&2.574\\0&0&0&1&|&-21.4661 \end{array}\right]$

R₁ → R₁ - 0.1582×R₄

$\left[\begin{array}{cccccc}1&0&0&0&|&2.5763\\0&1&0&0.5816&|&2.8112\\0&0&1&0.5153&|&2.574\\0&0&0&1&|&-21.4661\end{array}\right]$

R₂ → R₂ - 0.5816×R₄

$\left[\begin{array}{cccccc}1&0&0&0&|&2.5763\\0&1&0&0&|&15.2966\\0&0&1&0.5153&|&2.574\\0&0&0&1&|&-21.4661\end{array}\right]$

R₃ → R₃ - 0.5153×R₄

$\left[\begin{array}{cccccc}1&0&0&0&|&2.5763\\0&1&0&0&|&15.2966\\0&0&1&0&|&13.6356 \\0&0&0&1&|&-21.4661\end{array}\right]$

So,

u = 2.5763  

x = 15.2966  

y = 13.6356  

z = -21.4661

Answer 2

Answer:

Step-by-step explanation: