Math Problem Statement
Find the general solution of the system of linear equations in parametric vector form
Solution
The system of linear equations shown in the image is:
Let me solve this step by step and express the general solution in parametric vector form. sympy import symbols, Eq, solve, Matrix
Define the variables
x1, x2, x3, x4 = symbols('x1 x2 x3 x4')
Define the equations
eq1 = Eq(x1 + x2 + x3 - x4, 4) eq2 = Eq(3x1 + 2x2 + 3x3 + 4x4, 2) eq3 = Eq(4x1 + 3x2 + 5x3 - 5x4, 0) eq4 = Eq(3x1 + 4x2 + 6x3 + 6x4, -3)
Solve the system of equations
coeff_matrix = Matrix([[1, 1, 1, -1, 4], [3, 2, 3, 4, 2], [4, 3, 5, -5, 0], [3, 4, 6, 6, -3]])
Row reduce the augmented matrix to reduced row-echelon form (RREF)
rref_matrix, pivot_columns = coeff_matrix.rref() rref_matrixThe reduced row-echelon form (RREF) of the augmented matrix is:
1 & 0 & 0 & 0 & \frac{49}{20} \\ 0 & 1 & 0 & 0 & \frac{351}{40} \\ 0 & 0 & 1 & 0 & -\frac{37}{5} \\ 0 & 0 & 0 & 1 & -\frac{7}{40} \end{bmatrix}$$ This indicates that the system has a unique solution, as each variable $$x_1, x_2, x_3, x_4$$ is fully determined: $$x_1 = \frac{49}{20}, \quad x_2 = \frac{351}{40}, \quad x_3 = -\frac{37}{5}, \quad x_4 = -\frac{7}{40}.$$ Thus, the solution to the system of equations is: $$\mathbf{x} = \begin{bmatrix} \frac{49}{20} \\ \frac{351}{40} \\ -\frac{37}{5} \\ -\frac{7}{40} \end{bmatrix}.$$ Would you like a detailed explanation of how this solution was derived or have any further questions? --- Here are 5 related questions for exploration: 1. How do you verify if a solution satisfies all the original equations? 2. What are the steps to transform a matrix into reduced row-echelon form? 3. How can parametric solutions be found if the system is underdetermined? 4. How do pivots in a matrix correspond to leading variables in a solution? 5. What are the geometric interpretations of solutions to systems of linear equations? **Tip:** Always check for the consistency of a system of equations before solving!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Reduced Row-Echelon Form (RREF)
Matrix Operations
Formulas
Ax = b (system of linear equations)
Row reduction to RREF
Solution in vector form
Theorems
Fundamental Theorem of Linear Algebra
Suitable Grade Level
Undergraduate Level (Grades 12+)