2. Solve the system of equations by Gauss Elimination method 10x-2y+3z=23; 2x+10y-5z=-33; 3x-4y+10z=41. without decimal values

To solve the system of equations using the Gauss Elimination method, we'll proceed step by step, performing row operations to transform the system into an upper triangular form, and then apply back substitution to find the values of \( x \), \( y \), and \( z \).

The system of equations is:
\[
\begin{aligned}
1.\ & 10x - 2y + 3z = 23 \quad \text{(Equation 1)} \\
2.\ & 2x + 10y - 5z = -33 \quad \text{(Equation 2)} \\
3.\ & 3x - 4y + 10z = 41 \quad \text{(Equation 3)}
\end{aligned}
\]

### Step 1: Convert the system into an augmented matrix
The augmented matrix for the system is:

\[
\begin{pmatrix}
10 & -2 & 3 & | & 23 \\
2 & 10 & -5 & | & -33 \\
3 & -4 & 10 & | & 41
\end{pmatrix}
\]

### Step 2: Perform row operations to achieve an upper triangular form

#### Eliminate the first element in the second and third rows

1. **Row 2**: Subtract \( \frac{1}{5} \) of Row 1 from Row 2 to eliminate the first element of Row 2.
   \[
   R2 \rightarrow R2 - \frac{2}{10}R1 \Rightarrow R2 \rightarrow R2 - \frac{1}{5}R1
   \]

   \[
   \begin{aligned}
   R2 = & (2 \ - 2) \ 10 - \left(\frac{1}{5}(-2)\right) - 5 - \left(\frac{1}{5}3\right) \ - 33- \left(\frac{1}{5}\right)
   
   \]


Learn how to solve the system of equations 10x - 2y + 3z = 23, 2x + 10y - 5z = -33, 3x - 4y + 10z = 41 using the Gauss Elimination method. Step-by-step solution and detailed matrix operations included.

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