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Chapter 3: Problem 49

Translate the given systems of equations into matrix form. \(x-y=4\) \(2 x-y=0\)

Short Answer

Expert verified
The given system of equations can be represented in matrix form as: \[ \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \end{bmatrix} \]

Step by step solution

01

Identify the coefficients and constants

Rewrite the given system of equations, indicating the coefficients for each variable (x, y) and the constants: \(x - y = 4\) \(2x - y = 0\) The coefficients and constants for this system are: 1. Equation 1: Coefficients for x and y are (1, -1), and the constant term is 4. 2. Equation 2: Coefficients for x and y are (2, -1), and the constant term is 0.
02

Write the augmented matrix

Now, we will represent the system of equations as an augmented matrix, where the first column corresponds to the coefficients of x, the second column corresponds to the coefficients of y, and the third column corresponds to the constants. The augmented matrix for the given system is: \( \left[ \begin{array}{cc|c} 1 & -1 & 4 \\ 2 & -1 & 0 \end{array} \right] \)
03

Separate the matrix into A and B

Finally, separate the augmented matrix (A|B) into matrix A, containing the coefficients of the variables x and y, and matrix B, containing the constants: Matrix A: \[ A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix} \] Matrix B: \[ B = \begin{bmatrix} 4 \\ 0 \end{bmatrix} \] So, the given system of equations is represented in matrix form as \(A\vec{x} = \vec{B}\) : \[ \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \end{bmatrix} \]

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