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The third question involves testing the consistency and solving the system of linear equations. Let's break down and solve the two systems of equations provided in **3(i)** and **3(ii)**:

### Question 3(i) - Solving the system of linear equations

#### (a) Solve the system of equations:
1. \( 5x + 3y + 7z = 4 \)
2. \( 8x + 26y + 2z = 9 \)
3. \( 7x + 2y + 10z = 5 \)

To solve this system, we can use **Gaussian elimination** or **Cramer's Rule**. Let's proceed with Gaussian elimination:

1. Write the augmented matrix:
\[
\begin{pmatrix}
5 & 3 & 7 & | & 4 \\
8 & 26 & 2 & | & 9 \\
7 & 2 & 10 & | & 5 \\
\end{pmatrix}
\]

2. Apply **row operations** to transform it into **Row Echelon Form** (REF).

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#### (b) Solve the system of equations:
1. \( x + 2y - z = 3 \)
2. \( 3x - y + 2z = 1 \)
3. \( 2x - 2y + 3z = 2 \)
4. \( -x - y + z = -1 \)

Similarly, we will use **Gaussian elimination** or another method like **substitution** to solve this system.

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I will now begin solving these two systems of equations using the Gaussian elimination method.### Solutions for Question 3(i):

#### (a) The system of equations:
1. \( 5x + 3y + 7z = 4 \)
2. \( 8x + 26y + 2z = 9 \)
3. \( 7x + 2y + 10z = 5 \)

The solution is:
\[
x = 0, \quad y = \frac{5}{16}, \quad z = \frac{7}{16}
\]

#### (b) The system of equations:
1. \( x + 2y - z = 3 \)
2. \( 3x - y + 2z = 1 \)
3. \( 2x - 2y + 3z = 2 \)
4. \( -x - y + z = -1 \)

This system has **no solution**. The equations are inconsistent, meaning the system is over-determined, and no set of values for \(x\), \(y\), and \(z\) satisfies all the equations simultaneously.

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Would you like a detailed explanation of the Gaussian elimination steps or have any other questions?

### Related Questions:
1. How do you identify an inconsistent system of equations?
2. Can a system with more equations than unknowns have a solution?
3. What are the different methods for solving a system of linear equations?
4. How can you represent systems of equations geometrically?
5. What role does the determinant play in determining the uniqueness of a solution?

**Tip:** For systems of linear equations, check for consistency by transforming them into row echelon form. If you find a row of zeros in the coefficient matrix, the system may be inconsistent.

Solve the system of equations in problem 3(i) and discuss the values of λ, μ for consistency in 3(ii).

This page explains how to solve systems of linear equations using Gaussian elimination and tests for consistency in linear systems. It covers key concepts like row echelon form and examines the conditions for unique, infinite, or no solutions.

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