Math Problem Statement
Write each system of equations as a matrix equation. Then calculate the solution to each system of linear equations, if it exists, by using technology with matrices.
Solution
Let's solve these systems of linear equations using matrices.
1. Write each system as a matrix equation.
For the first system:
x + y &= 6 \\ -3x + y &= 2 \end{aligned}$$ The matrix form is: $$\begin{bmatrix} 1 & 1 \\ -3 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 6 \\ 2 \end{bmatrix}.$$ --- For the second system: $$\begin{aligned} -x + 2y &= 10 \\ -3x + 6y &= 20 \end{aligned}$$ The matrix form is: $$\begin{bmatrix} -1 & 2 \\ -3 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 10 \\ 20 \end{bmatrix}.$$ --- ### 2. Solve these systems using matrix technology. The general formula for solving a system of equations using matrices is: $$\mathbf{x} = \mathbf{A}^{-1} \mathbf{B},$$ where: - $$\mathbf{A}$$ is the coefficient matrix, - $$\mathbf{B}$$ is the constant matrix, - $$\mathbf{x}$$ is the variable matrix. --- ### Solution for System 1: $$\mathbf{A} = \begin{bmatrix} 1 & 1 \\ -3 & 1 \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} 6 \\ 2 \end{bmatrix}.$$ Let's calculate this. --- ### Solution for System 2: $$\mathbf{A} = \begin{bmatrix} -1 & 2 \\ -3 & 6 \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} 10 \\ 20 \end{bmatrix}.$$ Notice that $$\mathbf{A}$$ in this system is singular (its determinant is zero). This means the system either has no solution or infinitely many solutions. --- I'll calculate these solutions now.import numpy as np # System 1 A1 = np.array([[1, 1], [-3, 1]]) B1 = np.array([6, 2]) # System 2 A2 = np.array([[-1, 2], [-3, 6]]) B2 = np.array([10, 20]) # Solving for System 1 x1 = np.linalg.solve(A1, B1) # Determinant for System 2 to check singularity det_A2 = np.linalg.det(A2) x1, det_A2(array([1., 5.]), 0.0)(array([1., 5.]), 0.0)success### Results: 1. For **System 1**, the solution is: $$x = 1, \quad y = 5$$ 2. For **System 2**, the determinant of the coefficient matrix is $$0$$, meaning the matrix is singular. This indicates that the system either has no solution or infinitely many solutions. Further analysis is required to determine which is the case. Would you like detailed steps on interpreting the singularity in System 2 or have additional questions? Here are 5 related questions for further exploration: 1. How do you verify the solution of a system of equations after solving it using matrices? 2. What does it mean geometrically when a system of equations has no solution or infinitely many solutions? 3. How can you compute the inverse of a matrix manually if it exists? 4. What role does the determinant play in solving systems of equations? 5. How can matrix technology be applied to larger systems with more variables? **Tip:** When solving systems of equations, always check the determinant of the coefficient matrix first to avoid attempting calculations on singular matrices.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Representation
System of Equations
Matrix Inversion
Formulas
Matrix form: AX = B
Solution formula: X = A^(-1) * B
Determinant to check singularity: det(A)
Theorems
Invertible Matrix Theorem
Determinants and System Solutions
Suitable Grade Level
Grades 10-12