Math Problem Statement
I want to solve a matrix with parameters, basically I need to find when it has no solutions, when it has a single one and when it has an infinite amount of them, when it does I need to find the general solution. The matrix is 4 8 7 3c 3b, 1 2 2 c b, 2 4 2 (c-1) b
Solution
To solve the system of linear equations represented by the matrix with parameters, we'll analyze the augmented matrix and determine the conditions under which the system has no solution, a unique solution, or infinitely many solutions. Here is the given matrix:
4 & 8 & 7 & 3c & 3b \\ 1 & 2 & 2 & c & b \\ 2 & 4 & 2 & (c-1) & b \end{pmatrix}$$ Let's write down the corresponding system of equations: 1. $$4x + 8y + 7z = 3c$$ 2. $$x + 2y + 2z = c$$ 3. $$2x + 4y + 2z = c - 1$$ To find the solutions, we need to reduce the augmented matrix to row-echelon form (REF) or reduced row-echelon form (RREF). Let's start by representing the augmented matrix: $$\left[\begin{array}{ccc|c} 4 & 8 & 7 & 3c \\ 1 & 2 & 2 & c \\ 2 & 4 & 2 & c-1 \end{array}\right]$$ ### Step 1: Row Operations 1. Divide the first row by 4: $$\left[\begin{array}{ccc|c} 1 & 2 & 1.75 & 0.75c \\ 1 & 2 & 2 & c \\ 2 & 4 & 2 & c-1 \end{array}\right]$$ 2. Subtract the first row from the second row: $$\left[\begin{array}{ccc|c} 1 & 2 & 1.75 & 0.75c \\ 0 & 0 & 0.25 & 0.25c \\ 2 & 4 & 2 & c-1 \end{array}\right]$$ 3. Subtract 2 times the first row from the third row: $$\left[\begin{array}{ccc|c} 1 & 2 & 1.75 & 0.75c \\ 0 & 0 & 0.25 & 0.25c \\ 0 & 0 & -1.5 & -0.5c-1 \end{array}\right]$$ ### Step 2: Analyze the Reduced Matrix Let's focus on the second and third rows: $$\left[\begin{array}{ccc|c} 0 & 0 & 0.25 & 0.25c \\ 0 & 0 & -1.5 & -0.5c-1 \end{array}\right]$$ Divide the second row by 0.25 and the third row by -1.5: $$\left[\begin{array}{ccc|c} 0 & 0 & 1 & c \\ 0 & 0 & 1 & \frac{1 + 0.5c}{1.5} \end{array}\right]$$ $$\left[\begin{array}{ccc|c} 0 & 0 & 1 & c \\ 0 & 0 & 1 & \frac{1 + 0.5c}{-1.5} \end{array}\right]$$ Since the last two rows are inconsistent for any value of $$c$$, the system has no solution. ### General Solution Analysis Since the last two rows in the augmented matrix are inconsistent, it indicates that there is no value of $$c$$ or $$b$$ that can make the system consistent. Hence, the system has no solution under the given circumstances. Would you like more details or have any questions? Here are 5 numbered questions you might want to ask: 1. Can you explain how to find the row-echelon form (REF) of a matrix? 2. How do I interpret the row-echelon form to determine the number of solutions? 3. What are the next steps if the system had a solution? 4. How do I find the general solution for a system with infinitely many solutions? 5. Can you demonstrate a specific example where the system has a unique solution? **Tip:** When dealing with systems of linear equations, always check the consistency of the system by looking for rows in the augmented matrix that suggest contradictions, such as having a row with all zeros on the left and a non-zero entry on the right.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Augmented Matrix
Row Reduction
Formulas
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Theorems
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Suitable Grade Level
Advanced High School
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