Question

Using Gauss elimination and back substitution, solve

$\left[\begin{array}{ccc}8& 2& 1\\ 4& 6& 2\\ 3& 4& 14\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]\mathbf{=}\left[\begin{array}{c}3\\ 4\\ 2\end{array}\right]$

Short answer

Therefore, the solution is${\mathrm{x}}_1=0.182,{\mathrm{x}}_2=0.3202,\mathrm{and}{\mathrm{x}}_3=\frac{80}{89}.$

Step-by-step solution

Given data.

The given matrix is,

$\left[\begin{array}{ccc}8& 2& 1\\ 4& 6& 2\\ 3& 4& 14\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}3\\ 4\\ 2\end{array}\right]......\left(1\right)$

Determine the formulas of linear equation.

Write the formula of general form of linear equation.

$\mathbf{Ax}\mathbf{=}\mathbf{B}$ ……. (2)

Here, x is the$N\times 1$matrix which is the required solution, A is a$N\times N$matrix

which represents the coefficients of x, and B is the$N\times 1$matrix.

Here, N is order of the system.

Apply the Gauss Elimination method.

For Gauss Siedel elimination method, $N-1$steps are needed.

From matrix (1), the order of the matrix is 3 , therefore,

$$\begin{gathered} N-1=3-1 \\ =2 \end{gathered}$$

Step 1: Now, in equation (1), divide first row by 2 and subtract from row 2 and

multiply first row by $\frac{3}{8}$and subtract from row 3 .

Step 2: Now, in equation (3), multiply second row by and subtract from row .

$\left[\begin{array}{ccc}8& 2& 1\\ 0& 5& 1\\ 0& \frac{13}{4}& 14\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}3\\ 2.5\\ 13.625\end{array}\right]......\left(3\right)$

Solve linear equations.

Now, from equation (3),

$$\begin{gathered} 13.35{\mathrm{x}}_3=12......\left(4\right) \\ 5{\mathrm{x}}_2+{\mathrm{x}}_3=2.5......\left(5\right) \\ 8{\mathrm{x}}_1+2{\mathrm{x}}_2+{\mathrm{x}}_3=3......\left(6\right) \end{gathered}$$

Solve equation (4)

$$\begin{gathered} x_3=\frac{12}{13.35} \\ =\frac{80}{89} \end{gathered}$$

Substitute $\frac{80}{89}$for $x_3$in equation (5).

$$\begin{gathered} 5x_2+\frac{80}{89}=2.5 \\ {x}_2=0.3202 \end{gathered}$$

Substitute 0.3202 for role="math" localid="1655283112420" $x_2$and $\frac{80}{89}$for $x_3$in equation (6).

$$\begin{gathered} 8x_1+2\left(0.3202\right)+\frac{80}{89}=3 \\ {x}_1=0.182 \end{gathered}$$

Therefore, the solution is$x_1=0.182,x_2=0.3202,and{x}_3=\frac{80}{89}.$