Problem 5 - Algebra A Combined Function

Question

Use matrices to solve each system of linear equations. See Example 2. $$ \left\\{\begin{array}{r} x-2 y=4 \\ 2 x-4 y=4 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has infinitely many solutions since the determinant is zero.

Step 1: Write the System in Matrix Form

The given system of equations can be written in the matrix form as \(A\mathbf{x} = \mathbf{b}\), where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the column matrix of variables, and \(\mathbf{b}\) is the constant terms matrix. The system is: \(\begin{bmatrix} 1 & -2 \ 2 & -4 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 4 \ 4 \end{bmatrix}\).

Step 2: Find the Determinant of Coefficient Matrix

To solve the system using matrices, first compute the determinant of the coefficient matrix \(A\). The determinant, \(\text{det}(A)\), is calculated as follows: \(\text{det}(A) = 1(-4) - (-2)(2) = -4 + 4 = 0\).

Step 3: Check for Solvability

Since the determinant of the coefficient matrix is zero, the matrix is singular, meaning the system of equations either has no solutions or an infinite number of solutions. Analyze by inspection.

Step 4: Assess the System for Consistency

Check if the system is consistent. Since both equations simplify to \(x - 2y = 4\), they are effectively the same, implying the system has infinitely many solutions.

Key Concepts

Solving Systems of Linear Equations

When faced with a system of linear equations, the goal is often to find values for each variable that satisfy all the given equations simultaneously. Matrices can be an efficient tool for solving these systems. To solve a system using matrices, you convert the equations into matrix form, having a coefficient matrix that multiplies the variable matrix and equates to a constant matrix. This structured form makes it easier to perform operations and analyze the system. In the given example, the system of equations is transformed into a matrix equation. This technique helps in recognizing patterns and relationships among variables, simplifying the process of solving complex systems.

Coefficient Matrix

The coefficient matrix is a compact way of representing the system of equations using numbers. Each entry in this matrix corresponds to the coefficients of the variables in the equations.The matrix for this particular system is \[\begin{bmatrix} 1 & -2 \ 2 & -4 \end{bmatrix} \].- **Rows and Columns**: Each row of this matrix represents an equation, and each column represents the coefficients of a specific variable across the equations. These positions are vital for understanding variable interactions.- **Importance**: The coefficient matrix serves as a foundation for many mathematical operations, such as calculating determinants, which help determine the nature of the solutions of the system.

Determinant of a Matrix

The determinant is a special number that can be calculated from a square matrix. It provides important information about the system of equations, such as whether it has a unique solution, infinitely many solutions, or no solution at all.In our example, the determinant of the coefficient matrix \(\text{det}(A) = 1(-4) - (-2)(2) = 0\). - **Zero Determinant**: A determinant of zero indicates that the matrix is singular. This suggests that the equations might not be independent of each other, leading to either no solutions or infinitely many solutions.- **Role in Solving**: The determinant is a critical factor in determining solvability. If it's zero, other methods, like row reduction or substitution, must be used to assess the consistency of the system.

Infinitely Many Solutions

A system of equations can have infinitely many solutions if there are fewer unique equations than there are variables, which makes some equations redundant.In the provided system, each equation simplifies down to \(x - 2y = 4\),indicating they’re essentially the same equation. This identical nature points towards an infinite set of solutions, allowing for multiple values of \(x\) and \(y\) to satisfy the system.- **Characteristics**: Such systems often demonstrate dependent equations where multiple combinations of variable values result in true statements.- **Understanding**: Recognizing these characteristics assesses the feasibility of typical algebraic techniques like substitution or elimination. This leads one to conclude that infinitely many solutions exist, requiring alternative means to express solutions, such as parametric solutions.