Problem 39 - Precalculus Mathematics for Calculus

Question

Write the system of equations as a matrix equation. $$\left\\{\begin{array}{l} 2 x-5 y=7 \\ 3 x+2 y=4 \end{array}\right.$$

Step-by-Step Solution

Answer

The matrix equation is \[ \begin{bmatrix} 2 & -5 \\ 3 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ 4 \end{bmatrix}. \]

Step 1: Identify Coefficients and Constants

First, identify the coefficients of the variables and the constants from each equation. For the equation \(2x - 5y = 7\), the coefficients are 2 and -5, with 7 as the constant. For the equation \(3x + 2y = 4\), the coefficients are 3 and 2, with 4 as the constant.

Step 2: Form the Coefficient Matrix

Arrange the coefficients of \(x\) and \(y\) into a matrix. The first row will contain the coefficients from the first equation, and the second row will contain the coefficients from the second equation. Hence, the coefficient matrix is: \[ A = \begin{bmatrix} 2 & -5 \ 3 & 2 \end{bmatrix} \]

Step 3: Form the Variable Vector

Create a column vector for the variables \(x\) and \(y\). This vector will represent the variables of the system in the matrix equation:\[ X = \begin{bmatrix} x \ y \end{bmatrix} \]

Step 4: Form the Constant Vector

Write the constants from the right-hand side of the equations as a column vector:\[ B = \begin{bmatrix} 7 \ 4 \end{bmatrix} \]

Step 5: Combine into Matrix Equation

Combine the coefficient matrix, variable vector, and constant vector into a single matrix equation:\[ AX = B \]Thus, the system of equations can be represented as:\[ \begin{bmatrix} 2 & -5 \ 3 & 2 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 7 \ 4 \end{bmatrix} \]

Key Concepts

systems of equations

Systems of equations comprise multiple equations that share common variables. These systems are often used when trying to find values that satisfy all included equations. For instance, in a system with two equations, we typically look for solutions for two variables, such as the values of \(x\) and \(y\) that make both equations true simultaneously.
Here is how a typical system might look:

• Equation 1: \( 2x - 5y = 7 \)
• Equation 2: \( 3x + 2y = 4 \)

To solve such a system, one might use various methods like substitution, elimination, or transforming the system into a matrix equation, which can then be solved with matrix operations.

coefficient matrix

The coefficient matrix plays a crucial role when converting a system of equations into a matrix equation. This matrix consists solely of the coefficients from the variables in each equation, organized in a structured form.
Let's consider the equations we have:

• For the first equation \(2x - 5y = 7\), the coefficients are 2 and -5.
• For the second equation \(3x + 2y = 4\), the coefficients are 3 and 2.

Once extracted, these coefficients are arranged into a matrix like this:\[A = \begin{bmatrix} 2 & -5 \ 3 & 2 \end{bmatrix}\]This coefficient matrix, \(A\), is essential for solving the system using matrix algebra, enabling us to automatically include the impact of these coefficients in subsequent calculations.

variable vector

A variable vector represents the variables involved in our system of equations, bundled together into a matrix form. It simplifies the way these variables are handled in a matrix equation configuration.
In our system, the variables are \(x\) and \(y\). Therefore, they form this column vector:\[X = \begin{bmatrix} x \ y \end{bmatrix}\]The variable vector \(X\) operates alongside the coefficient matrix and the constant vector to reflect how changes in the variables will directly affect the outcomes represented by the equations.

constant vector

The constant vector in a system of equations contains the constants from the right-hand side of each equation. These constants are the values that stand alone, not attached to any variable in the equations.
From our example system:

• The constant from the first equation \(2x - 5y = 7\) is 7.
• The constant from the second equation \(3x + 2y = 4\) is 4.

These constants are arranged into a column vector:\[B = \begin{bmatrix} 7 \ 4 \end{bmatrix}\]This constant vector \(B\) ensures that the isolated constants are correctly represented within the matrix equation, critical for solving the system by balancing the equation on both sides.