2. Solve the system of equations by Gauss Elimination method 10x-2y+3=23; 2x+10y-5=-33; 3x-4y+10z=41. upper triangular matrix must be include

To solve the given system of equations using the Gauss Elimination method, let's first rewrite the equations in a standard form:

1. \( 10x - 2y + 0z = 20 \) (after subtracting 3 from both sides)
2. \( 2x + 10y + 0z = -28 \) (after adding 5 to both sides)
3. \( 3x - 4y + 10z = 41 \)

We will now represent the system in an augmented matrix form:

\[
\begin{pmatrix}
10 & -2 & 0 & \vert & 20 \\
2 & 10 & 0 & \vert & -28 \\
3 & -4 & 10 & \vert & 41
\end{pmatrix}
\]

### Step 1: Convert to Upper Triangular Matrix
We'll perform row operations to convert this matrix into an upper triangular form.

#### First Operation: Eliminate \(x\) from the second and third rows
- **Row 2:** \( R2 = R2 - \frac{2}{10} R1 = R2 - 0.2 R1 \)
- **Row 3:** \( R3 = R3 - \frac{3}{10} R1 = R3 - 0.3 R1 \)

After performing these operations:

\[
\begin{pmatrix}
10 & -2 & 0 & \vert & 20 \\
0 & 10.4 & 0 & \vert & -32 \\
0 & -3.4 & 10 & \vert & 35
\end{pmatrix}
\]

#### Second Operation: Eliminate \(y\) from the third row
- **Row 3:** \( R3 = R3 - \frac{-3.4}{10.4} R2 = R3 + 0.327 R2 \)

After performing this operation:

\[
\begin{pmatrix}
10 & -2 & 0 & \vert & 20 \\
0 & 10.4 & 0 & \vert & -32 \\
0 & 0 & 10 & \vert & 24.544
\end{pmatrix}
\]

Now, the matrix is in upper triangular form.

### Step 2: Back Substitution

We can now solve for the variables using back substitution:

1. From Row 3: \( 10z = 24.544 \) 
   \[
   z = \frac{24.544}{10} = 2.4544
   \]

2. From Row 2: \( 10.4y = -32 \) 
   \[
   y = \frac{-32}{10.4} = -3.0769
   \]

3. From Row 1: \( 10x - 2y = 20 \)
   \[
   10x = 20 + 2(3.0769) = 20 + 6.1538 = 26.1538
   \]
   \[
   x = \frac{26.1538}{10} = 2.6154
   \]

### Final Solution
The solution to the system of equations is:

\[
x = 2.6154, \quad y = -3.0769, \quad z = 2.4544
\]

Would you like more details or have any questions?

### Related Questions
1. How do you apply the Gauss-Jordan elimination method to solve a system of equations?
2. What are the differences between Gauss Elimination and Gauss-Jordan Elimination methods?
3. How can you solve a system of equations using matrix inversion?
4. What are the advantages of using the Gauss Elimination method in linear algebra?
5. Can you explain how pivoting affects the stability of the Gauss Elimination method?

### Tip
Always perform a check by substituting the values of \(x\), \(y\), and \(z\) back into the original equations to ensure the solution is correct.

Learn how to solve a system of equations using the Gauss Elimination method with upper triangular matrix. This problem involves transforming the system into an upper triangular matrix and performing back substitution to find the solutions for x, y, and z. Suitable for undergraduate level students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation