Question

Solve the system of equations
7 3x + 2y + 4z =
7 2x + y + z =
2 x + 3y + 5z =
using Gauss elimination with pivoting. Store the multipliers and also write the
pivoting vector.

Answer 1

Answer on Question #46677 – Math – Algorithms | Quantitative Methods

Solve the system of equations

$$7\ 3x + 2y + 4z =$$ $$7\ 2x + y + z =$$ $$2x + 3y + 5z =$$

using Gauss elimination with pivoting. Store the multipliers and also write the pivoting vector.

Solution.

$$\left\{ \begin{array}{c} 3x + 2y + 4z = 7 \\ 2x + y + z = 7 \\ x + 3y + 5z = 2 \end{array} \right.$$

- Swap Row 1 and Row 3. After this step we have:

$$\left\{ \begin{array}{c} x + 3y + 5z = 2 \\ 2x + y + z = 7 \\ 3x + 2y + 4z = 7 \end{array} \right.$$

- Multiply the first equation by $-2$ and add the result to the second equation. The result is:

$$\left\{ \begin{array}{c} x + 3y + 5z = 2 \\ -5y - 9z = 3 \\ 3x + 2y + 4z = 7 \end{array} \right.$$

- Multiply the first equation by $-3$ and add the result to the third equation. The result is:

$$\left\{ \begin{array}{c} x + 3y + 5z = 2 \\ -5y - 9z = 3 \\ -7 - 11z = 1 \end{array} \right.$$

- Multiply the second equation by $-\frac{7}{5}$ and add the result to the third equation. The result is:

$$\left\{ \begin{array}{c} x + 3 y + 5 z = 2 \\ - 5 y - 9 z = 3 \\ \frac {8}{5} z = - \frac {16}{5} \end{array} \right.$$

- Solve for z.

$$z = - 2$$

- Solve for y.

$$- 5 y - 9 z = 3 \rightarrow y = 3$$

- solve for $x$ by substituting $y = 3$ and $z = -2$ into the first equation.

$$x + 3 y + 5 z = 2 \rightarrow x = 3.$$

Finally, $x = 3$, $y = 3$, $z = -2$.

www.AssignmentExpert.com