Math Problem Statement
đáp án
Tìm m để hệ phương trình tuyến tính có vô số nghiệm.
Solution
The problem asks for the value of such that the system of linear equations has infinitely many solutions. The system of equations is given as:
2x + 2y - 4z = m \\ -3x + 5y - z = 3 \\ -4x - 4y + 8z = -2 \end{cases}$$ ### Step 1: Convert the system into a matrix We will solve this system using the concept of linear dependence and the rank of the matrix. For the system to have infinitely many solutions, the equations must be linearly dependent, and the augmented matrix should have a rank equal to the number of variables minus 1. The coefficient matrix is: $$\begin{pmatrix} 2 & 2 & -4 \\ -3 & 5 & -1 \\ -4 & -4 & 8 \end{pmatrix}$$ The augmented matrix (including the constants on the right-hand side) is: $$\begin{pmatrix} 2 & 2 & -4 & m \\ -3 & 5 & -1 & 3 \\ -4 & -4 & 8 & -2 \end{pmatrix}$$ ### Step 2: Apply the condition for infinite solutions For the system to have infinitely many solutions, the determinant of the coefficient matrix must be zero, indicating that the equations are linearly dependent. Let's calculate the determinant of the coefficient matrix. $$\text{Determinant} = \begin{vmatrix} 2 & 2 & -4 \\ -3 & 5 & -1 \\ -4 & -4 & 8 \end{vmatrix}$$ Let's calculate this determinant.The determinant of the coefficient matrix is $$0$$. This confirms that the system of equations is linearly dependent, meaning it has either no solutions or infinitely many solutions. ### Step 3: Check consistency of the system To determine the value of $$m$$ that leads to infinitely many solutions, we need to check the consistency of the augmented matrix. Let's simplify the augmented matrix to check the condition. $$\begin{pmatrix} 2 & 2 & -4 & m \\ -3 & 5 & -1 & 3 \\ -4 & -4 & 8 & -2 \end{pmatrix}$$ We'll solve this matrix step by step to find the value of $$m$$.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can guide you through the manual steps needed to find the correct value of $$m$$. Would you prefer that? Here are some related questions: 1. How do you determine when a system of linear equations has infinitely many solutions? 2. What are the necessary conditions for a matrix to be consistent? 3. Can you solve a system of equations using the inverse of a matrix? 4. How does the rank of a matrix relate to the solution of a system of equations? 5. What does it mean when the determinant of a matrix is zero? **Tip**: When solving a system of equations, always check the determinant of the coefficient matrix to understand the system's solvability.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Systems
Matrix Determinants
Infinite Solutions
Formulas
Determinant of a 3x3 matrix
Conditions for infinite solutions
Theorems
Rank-Nullity Theorem
Consistency of Linear Systems
Suitable Grade Level
Grades 10-12
Related Recommendation
Analyzing the Linear System for Infinite Solutions Using Matrix Rank
Finding the Value of m for Infinite Solutions in Simultaneous Equations
How Many Solutions Does This System of Linear Equations Have?
Understanding Systems of Linear Equations and Infinite Solutions
Conditions for Solutions in a System of Linear Equations