Math Problem Statement
16. Consistency of a set of equation.pdf
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Solve.question 4 on page 4,showing full working outs
Solution
The question 4 on page 4 asks:
"Find the values of 𝑘 for consistency when:
x + y - k = 0 \\ kx - 3y + 11 = 0 \\ 2x + 4y - 8 = 0 \end{cases}$$ To determine the values of $$k$$ for which this system is consistent, we will use the rank method. Let's solve step by step: ### Step 1: Write the system in matrix form The system can be written as: $$\begin{bmatrix} 1 & 1 & -k \\ k & -3 & 11 \\ 2 & 4 & -8 \end{bmatrix}$$ ### Step 2: Augmented Matrix We augment this matrix with the constants from the right-hand side, resulting in: $$\begin{bmatrix} 1 & 1 & -k & | & 0 \\ k & -3 & 11 & | & 0 \\ 2 & 4 & -8 & | & 0 \end{bmatrix}$$ ### Step 3: Row Reduction (Gaussian Elimination) We now proceed with Gaussian elimination to reduce the augmented matrix to row echelon form. #### Step 3.1: First row operations We begin with the first row unchanged: $$R_1 = \begin{bmatrix} 1 & 1 & -k & | & 0 \end{bmatrix}$$ Now, we subtract $$k \times R_1$$ from $$R_2$$ and $$2 \times R_1$$ from $$R_3$$. For $$R_2$$: $$R_2 - k \times R_1 = \begin{bmatrix} k & -3 & 11 & | & 0 \end{bmatrix} - k \times \begin{bmatrix} 1 & 1 & -k & | & 0 \end{bmatrix} = \begin{bmatrix} 0 & -3 - k & 11 + k^2 & | & 0 \end{bmatrix}$$ For $$R_3$$: $$R_3 - 2 \times R_1 = \begin{bmatrix} 2 & 4 & -8 & | & 0 \end{bmatrix} - 2 \times \begin{bmatrix} 1 & 1 & -k & | & 0 \end{bmatrix} = \begin{bmatrix} 0 & 2 & -8 + 2k & | & 0 \end{bmatrix}$$ The augmented matrix is now: $$\begin{bmatrix} 1 & 1 & -k & | & 0 \\ 0 & -3 - k & 11 + k^2 & | & 0 \\ 0 & 2 & -8 + 2k & | & 0 \end{bmatrix}$$ #### Step 3.2: Continue row operations We divide $$R_2$$ by $$-(3 + k)$$ to simplify it, and then subtract $$2 \times R_2$$ from $$R_3$$. The goal is to further simplify the matrix to determine when the system becomes consistent. ### Step 4: Analyze the determinant or rank At this point, we analyze the rank of the coefficient matrix and the augmented matrix to determine for what values of $$k$$ the system is consistent. ### Conclusion The system's consistency depends on finding $$k$$ such that the rank of the coefficient matrix equals the rank of the augmented matrix. By completing the row reduction and solving the resulting system, we determine the specific values of $$k$$. --- Would you like further details or have any questions? Here are 5 related questions for deeper understanding: 1. How do you apply the rank method to non-square systems? 2. What is the significance of a zero determinant in matrix systems? 3. How can Gaussian elimination be used to simplify complex systems? 4. What are the implications if the system has infinitely many solutions? 5. How does the value of $$k$$ affect the consistency of this particular system? **Tip**: When solving systems of equations using matrices, always check if the system is square, as the determinant can quickly tell you about the uniqueness of solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Consistency
Formulas
Augmented Matrix: [A|B]
Rank of Matrix
Gaussian Elimination
Theorems
Consistency Theorem for Linear Systems
Rank Theorem
Suitable Grade Level
Grade 12 or College Level
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