Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}&3 x+2 y=-4\\\&2 x-y=-5\end{aligned}$$

Step-by-Step Solution

Answer

The solution to the system is \(x = -2\) and \(y = 1\).

Step 1: Write the System in Matrix Form

Express the system of equations in matrix form, which involves writing it in the form \(A\vec{x} = \vec{b}\). Here, the coefficient matrix \(A\), variable vector \(\vec{x}\), and constants vector \(\vec{b}\) are: \(A = \begin{pmatrix} 3 & 2 \ 2 & -1 \end{pmatrix}\), \(\vec{x} = \begin{pmatrix} x \ y \end{pmatrix}\), and \(\vec{b} = \begin{pmatrix} -4 \ -5 \end{pmatrix}\).

Step 2: Calculate Determinant of Coefficient Matrix (D)

Find the determinant \(D\) of the matrix \(A\). Use the formula \(D = a_{11}a_{22} - a_{12}a_{21}\) for a 2x2 matrix. For this matrix, \(D = (3)(-1) - (2)(2)\).

Step 3: Compute Determinant D

Calculate \(D = 3(-1) - 2(2) = -3 - 4 = -7\). Since \(D eq 0\), we can proceed with Cramer's rule.

Step 4: Determine Determinant for x (D_x)

Replace the first column of \(A\) with \(\vec{b}\), forming \(A_x = \begin{pmatrix} -4 & 2 \ -5 & -1 \end{pmatrix}\). Calculate \(D_x = (-4)(-1) - (2)(-5) = 4 + 10 = 14\).

Step 5: Calculate Determinant for y (D_y)

Replace the second column of \(A\) with \(\vec{b}\), forming \(A_y = \begin{pmatrix} 3 & -4 \ 2 & -5 \end{pmatrix}\). Calculate \(D_y = 3(-5) - (-4)(2) = -15 + 8 = -7\).

Step 6: Solve for Variables x and y

Using Cramer's rule, compute \(x = \frac{D_x}{D}\) and \(y = \frac{D_y}{D}\). Substituting the determinant values, \(x = \frac{14}{-7} = -2\) and \(y = \frac{-7}{-7} = 1\).

Key Concepts

System of Equations

A system of equations is a set of two or more equations that have common variables. In this context, we are dealing with linear equations, meaning each term is either a constant or the product of a constant and a variable. The goal is to find a common solution for the variables that satisfies all the given equations. For the exercise provided, we are working with the system:

• \(3x + 2y = -4\)
• \(2x - y = -5\)

To solve such a system, you can use several methods including substitution, elimination, or matrix methods like Cramer's Rule. Each method aims to find values of \(x\) and \(y\) that make both equations true simultaneously.

Determinant

The determinant is a special number that can be calculated from a square matrix, giving information about the matrix such as whether it is invertible. For a 2x2 matrix:\[A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\]the determinant is calculated as \(D = ad - bc\). In Cramer's Rule, knowing that \(D eq 0\) is essential, as it tells us the system of equations has a unique solution. If \(D = 0\), another method must be used. In our exercise, we computed the determinant of the coefficient matrix:\[D = 3(-1) - 2(2) = -3 - 4 = -7\]Since \(D eq 0\), we can proceed with Cramer's Rule to find the solutions for \(x\) and \(y\).

Matrix Form

Writing a system of equations in matrix form simplifies the process of using matrix operations, such as those required for Cramer's Rule. The system is expressed as:\[A\vec{x} = \vec{b}\]where \(A\) is the coefficient matrix, \(\vec{x}\) is the variable vector, and \(\vec{b}\) is the constant vector. In our exercise:

• The coefficient matrix \(A = \begin{pmatrix} 3 & 2 \ 2 & -1 \end{pmatrix}\)
• The variable vector \(\vec{x} = \begin{pmatrix} x \ y \end{pmatrix}\)
• The constant vector \(\vec{b} = \begin{pmatrix} -4 \ -5 \end{pmatrix}\)

This matrix form is crucial for applying methods like Cramer's Rule, which relies on matrix manipulations and determinants.

Variable Vector

The variable vector in matrix form represents the unknowns in the system of equations and is critical to solving systems using matrix operations. In our example, the variable vector is noted as:\[\vec{x} = \begin{pmatrix} x \ y \end{pmatrix}\]This compact representation holds all the variables of the system. By focusing on the variable vector, we allow matrix operations to handle the complexity of multiple variables systematically. When using Cramer's Rule, specifically, we replace columns in the coefficient matrix with the constant vector to find each variable's determinant:

• For \(x\), use \(A_x\), where the first column of \(A\) is replaced by \(\vec{b}\)
• For \(y\), use \(A_y\), where the second column of \(A\) is replaced by \(\vec{b}\)

Through this process, solving for each element of \(\vec{x}\) becomes a routine of computing determinants.