Find the solution of equation by gaussian elimination method 5x - 3y = 8, 10x – 6y = 16.

To solve the system of equations using the Gaussian elimination method, we'll convert the system into an augmented matrix and perform row operations to reach row-echelon form. Let's go through the steps:

### System of Equations

1. \(5x - 3y = 8\)
2. \(10x - 6y = 16\)

### Augmented Matrix

First, write the augmented matrix for the system:

\[
\begin{bmatrix}
5 & -3 & | & 8 \\
10 & -6 & | & 16
\end{bmatrix}
\]

### Step-by-Step Gaussian Elimination

#### Step 1: Row Operations

1. **R2 = R2 - 2 \cdot R1**

   To eliminate the first element of the second row, we subtract 2 times the first row from the second row:

\[
\begin{bmatrix}
5 & -3 & | & 8 \\
0 & 0 & | & 0
\end{bmatrix}
\]

### Interpretation

After performing the row operation, the second row of the matrix becomes zero:

\[
0x + 0y = 0
\]

This indicates that the second equation is a multiple of the first, meaning the system has infinitely many solutions (the two equations are dependent).

### Conclusion

The system of equations represents the same line in two different forms, which means there are infinitely many solutions along that line. Any point \((x, y)\) on the line described by the equation:

\[
5x - 3y = 8
\]

### Parametric Form of the Solution

To express the solution in parametric form, solve for \(y\) in terms of \(x\) using one of the equations:

From \(5x - 3y = 8\):

\[
3y = 5x - 8
\]

\[
y = \frac{5}{3}x - \frac{8}{3}
\]

Thus, the parametric solution is:

\[
(x, y) = \left(x, \frac{5}{3}x - \frac{8}{3}\right)
\]

Where \(x\) can be any real number. This describes all points on the line.

### Would You Like Further Details?

If you have questions or need more explanation, please feel free to ask!

### Further Questions

1. What is the difference between row-echelon form and reduced row-echelon form?
2. How can Gaussian elimination be applied to larger systems of equations?
3. What are some advantages of using Gaussian elimination?
4. Can Gaussian elimination be used for non-linear systems?
5. How do we determine if a system is consistent, inconsistent, or dependent using Gaussian elimination?
6. How does the rank of a matrix relate to the number of solutions?
7. What role do pivot elements play in Gaussian elimination?
8. How do you handle systems with no solutions in Gaussian elimination?

### Tip

When using Gaussian elimination, keep track of row operations carefully, as small arithmetic errors can lead to incorrect solutions.

Learn how to solve the system of equations 5x - 3y = 8 and 10x – 6y = 16 using the Gaussian elimination method. This problem covers key concepts in linear algebra, including systems of linear equations and Gaussian elimination. Suitable for advanced high school students and above.

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