Problem 22 - Intermediate Algebra

Question

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{l} 4 x-y=11 \\ 2 x+3 y=23 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution set is \((x, y) = (4, 5)\).

Step 1: Write the system in standard form

The given system of equations is already in standard form:1. \(4x - y = 11\)2. \(2x + 3y = 23\).

Step 2: Determine the coefficient matrix and find its determinant

The coefficient matrix \(A\) is \( \begin{pmatrix} 4 & -1 \ 2 & 3 \end{pmatrix} \).The determinant of \(A\) can be calculated as:\[ \text{det}(A) = 4 \times 3 - (-1) \times 2 = 12 + 2 = 14. \]

Step 3: Compute the determinant of the x-matrix

Replace the x-column of \(A\) with the constant terms to get the x-matrix:\( \begin{pmatrix} 11 & -1 \ 23 & 3 \end{pmatrix} \).The determinant of this matrix is:\[ \text{det}(A_x) = 11 \times 3 - (-1) \times 23 = 33 + 23 = 56. \]

Step 4: Compute the determinant of the y-matrix

Replace the y-column of \(A\) with the constant terms to get the y-matrix:\( \begin{pmatrix} 4 & 11 \ 2 & 23 \end{pmatrix} \).The determinant of this matrix is:\[ \text{det}(A_y) = 4 \times 23 - 11 \times 2 = 92 - 22 = 70. \]

Step 5: Apply Cramer's Rule to solve for \(x\) and \(y\)

Using Cramer's Rule, \( x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{56}{14} = 4 \) and\( y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{70}{14} = 5 \).

Key Concepts

Determinants

A determinant is a special number that can be calculated from a square matrix, which is pivotal when solving equations using Cramer's Rule. It provides important information about the matrix, such as whether it has an inverse. The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is calculated using the formula: \[ \text{det}(A) = ad - bc \] In our problem, the coefficient matrix is \( \begin{pmatrix} 4 & -1 \ 2 & 3 \end{pmatrix} \). By using the formula, we find the determinant to be \( 12 + 2 \), which equals 14. Determinants help us understand systems of equations better by showing if a unique solution exists. If the determinant is zero, the system might be dependent or inconsistent, meaning it doesn't have a single solution.

Coefficient Matrix

The coefficient matrix organizes the coefficients of the variables from a system of equations into a matrix form. This structured layout assists in applying various matrix solving techniques, like Cramer's Rule. For instance, in our equations:\[ 4x - y = 11 \]\[ 2x + 3y = 23 \]The coefficient matrix \( A \) is:\[ \begin{pmatrix} 4 & -1 \ 2 & 3 \end{pmatrix} \] Each row of the matrix corresponds to one equation, and each column corresponds to one variable. This format makes it easier to perform operations and calculate determinants.Using this coefficient matrix is essential when employing Cramer’s Rule, as it lays the foundation for replacing columns with constants to find the variables' determinants.

Systems of Equations

A system of equations is a collection of two or more equations with the same set of variables. The primary goal is to find values for these variables that satisfy all the equations simultaneously.In our given problem, the system consists of two equations:- \( 4x - y = 11 \)- \( 2x + 3y = 23 \)Systems can have:

• An unique solution
• No solution
• Infinitely many solutions

An essential technique for solving systems like these is Cramer’s Rule, which relies on determinants. By applying Cramer's Rule, we rearrange coefficients and constant terms into matrices that help compute the values of \( x \) and \( y \).This method is efficient for small systems, especially when the determinant of the coefficient matrix is not zero, ensuring a unique solution.