https://file.aiquickdraw.com/mathbot/file/1727274261750b0zimss5.jpg

The question asks to investigate the consistency of three sets of systems of linear equations and, if possible, find the solutions. I'll walk through each system to determine if it is consistent (has a solution) and, if so, solve it.

### (i) System of Equations:
\[
\begin{aligned}
2x + 3y + z &= 11 \\
x + y + z &= 6 \\
5x - y + 10z &= 34
\end{aligned}
\]

### (ii) System of Equations:
\[
\begin{aligned}
x - 5y + 3z &= -1 \\
2x - y - z &= 5 \\
5x - 7y + z &= 2
\end{aligned}
\]

### (iii) System of Equations:
\[
\begin{aligned}
4x - 2y + 6z &= 8 \\
x + y - 3z &= -1 \\
15x - 3y + 9z &= 21
\end{aligned}
\]

### Approach
To investigate the consistency and find solutions, we can:
1. Set up each system as a matrix (or augmented matrix for Gaussian elimination).
2. Use techniques like row reduction (Gaussian or Gauss-Jordan elimination), or Cramer's rule to determine consistency.
3. Solve if consistent.

Let me begin solving these systems step by step. I'll proceed with Gaussian elimination for each.Here are the results for each system of equations:

### (i) Solution:
The system has a unique solution:
\[
x = 3, \quad y = 1, \quad z = 2
\]

### (ii) System:
The solution involves very large numbers, suggesting that the system is either inconsistent or has an infinite number of solutions. This typically indicates that the system is either over-determined or has dependent equations.

### (iii) System:
The system is inconsistent, meaning there is no solution.

---

Would you like more details or explanations? Here are 5 related questions to deepen your understanding:
1. What does it mean when a system has infinite solutions?
2. How do we detect inconsistency in a system of linear equations?
3. What is Gaussian elimination, and how does it help in solving systems?
4. Can a system of equations have exactly two solutions? Why or why not?
5. How do you interpret the results when solving a system of linear equations using matrices?

**Tip:** When working with large matrices, checking for linear dependency (row operations) helps identify if a system has no or infinite solutions before solving.

Investigate for consistency of the following equations and if possible find the solutions: (i) 2x + 3y + z = 11, x + y + z = 6, 5x - y + 10z = 34, (ii) x - 5y + 3z = -1, 2x - y - z = 5, 5x - 7y + z = 2, (iii) 4x - 2y + 6z = 8, x + y - 3z = -1, 15x - 3y + 9z = 21

Investigate and solve systems of linear equations for consistency. The solution involves Gaussian elimination and matrix techniques to find solutions or determine inconsistency. Ideal for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation