Problem 34 - Precalculus Mathematics for Calculus

Question

Use Cramer's Rule to solve the system. $$\left\\{\begin{array}{l} 6 x+12 y=33 \\ 4 x+7 y=20 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution is \(x = 1.5\) and \(y = 2\).

Step 1: Understand Cramer's Rule

Cramer's Rule provides a method to solve a system of linear equations with as many equations as unknowns using determinants. For a 2x2 system, it involves calculating the determinants of matrices derived from the coefficient matrix.

Step 2: Write the Coefficient Matrix

The given system of equations is: 1. \(6x + 12y = 33\)2. \(4x + 7y = 20\)The coefficient matrix \(A\) is constructed using the coefficients of \(x\) and \(y\): \[ A = \begin{pmatrix} 6 & 12 \ 4 & 7 \end{pmatrix} \]

Step 3: Find Determinant of the Coefficient Matrix (det(A))

The determinant of matrix \(A\) is given by the formula: \[ \det(A) = 6 \times 7 - 12 \times 4 = 42 - 48 = -6 \]

Step 4: Construct the Matrices for Numerator Determinants

To find the value of \(x\), replace the first column of \(A\) with the constants from the right side of the equations. For \(y\), replace the second column:- Matrix \(A_x\) for \(x\):\[ A_x = \begin{pmatrix} 33 & 12 \ 20 & 7 \end{pmatrix} \] - Matrix \(A_y\) for \(y\):\[ A_y = \begin{pmatrix} 6 & 33 \ 4 & 20 \end{pmatrix} \]

Step 5: Calculate Determinants of A_x and A_y

Compute the determinant for each matrix:- \(\det(A_x) = 33 \times 7 - 12 \times 20 = 231 - 240 = -9\)- \(\det(A_y) = 6 \times 20 - 33 \times 4 = 120 - 132 = -12\)

Step 6: Find Values of x and y Using Cramer's Rule

According to Cramer's Rule, the solutions for \(x\) and \(y\) are given by:- \(x = \frac{\det(A_x)}{\det(A)} = \frac{-9}{-6} = 1.5\)- \(y = \frac{\det(A_y)}{\det(A)} = \frac{-12}{-6} = 2\)

Key Concepts

Determinants

Determinants play a crucial role in solving systems of linear equations using Cramer's Rule. Essentially, a determinant is a special number that can be calculated from a square matrix. In a 2x2 matrix, represented as \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated as \( ad - bc \).
To solve a system of equations, we need to compute the determinant of the coefficient matrix, referred to as \( \det(A) \). This determinant tells us whether the system has a unique solution.
- If \( \det(A) eq 0 \), the system has a unique solution.- If \( \det(A) = 0 \), the system may either have no solutions or infinitely many solutions.
In the example given, the determinant \( \det(A) = -6 \) is non-zero, indicating a unique solution exists. Calculating determinants quickly becomes second-nature when you apply them regularly to solve systems.

Systems of Linear Equations

A system of linear equations consists of two or more equations made up of two variables, which have a common solution. Each equation in the system represents a line on a graph, and the solution is represented by the intersection of these lines.
For a simple system of two variables \( x \) and \( y \), the algebraic representation might look like:

• \( ax + by = c \)
• \( dx + ey = f \)

The goal is to find the values of \( x \) and \( y \) that satisfy both equations when graphed, meaning they exist at the intersection point.
In our exercise, we have:

• \( 6x + 12y = 33 \)
• \( 4x + 7y = 20 \)

Using methods like substitution or elimination can solve this, but Cramer's Rule provides a unique determinant-based method, making it a preferred choice when dealing with multiple equations.

2x2 Matrices

Matrices are a systematic way of arranging numbers in rows and columns. A 2x2 matrix is the simplest form of a square matrix, having two rows and two columns, making calculations straightforward.
For example, a 2x2 matrix can be written as:\[\begin{pmatrix} a & b \ c & d \end{pmatrix}\]Matrices are helpful in representing and solving systems of linear equations. The coefficients of the variables in the equations are placed into this matrix form, known as the coefficient matrix, \( A \).
In the given problem, the coefficient matrix \( A \) of the equations is structured as:\[A = \begin{pmatrix} 6 & 12 \ 4 & 7 \end{pmatrix}\]Two derived matrices, \( A_x \) and \( A_y \), are created by replacing columns of \( A \) with the constant terms from the equations. They are instrumental in finding the variables using determinants. Understanding the basic setup of 2x2 matrices lays the foundation for dealing with larger systems.