Problem 34 - Algebra and Trigonometry

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: Write the Coefficient Matrix

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

Step 2: Compute the Determinant of the Coefficient Matrix

Calculate the determinant of the matrix \(A\). The formula for the determinant of a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) is \(ad - bc\).\[\text{Det}(A) = 6 \times 7 - 12 \times 4 = 42 - 48 = -6\]

Step 3: Write the Matrix for Variable x

To find \(x\), replace the first column of the coefficient matrix with the constants: \[A_x = \begin{pmatrix} 33 & 12 \ 20 & 7 \end{pmatrix}\]

Step 4: Compute the Determinant for Variable x

Calculate the determinant of matrix \(A_x\): \[\text{Det}(A_x) = 33 \times 7 - 12 \times 20 = 231 - 240 = -9\]

Step 5: Write the Matrix for Variable y

To find \(y\), replace the second column of the coefficient matrix with the constants: \[A_y = \begin{pmatrix} 6 & 33 \ 4 & 20 \end{pmatrix}\]

Step 6: Compute the Determinant for Variable y

Calculate the determinant of matrix \(A_y\): \[\text{Det}(A_y) = 6 \times 20 - 33 \times 4 = 120 - 132 = -12\]

Step 7: Solve for x and y Using Cramer's Rule

Cramer's Rule provides the solutions: \[ x = \frac{\text{Det}(A_x)}{\text{Det}(A)} = \frac{-9}{-6} = \frac{3}{2} = 1.5 \]\[ y = \frac{\text{Det}(A_y)}{\text{Det}(A)} = \frac{-12}{-6} = 2 \]

Key Concepts

Understanding Systems of Equations

A "system of equations" involves multiple equations that share common variables. In most cases, we wish to find out the values of these variables that satisfy all equations in the system at once.
In our exercise with Cramer’s Rule, the system includes two equations:

• Equation 1: \(6x + 12y = 33\)
• Equation 2: \(4x + 7y = 20\)

Here, "\(x\)" and "\(y\)" are the variables we need to solve for. Solving such a problem usually involves finding the point where the two lines represented by the equations intersect. In simpler systems like this one, we have exactly the same number of equations as variables, specifically two, making the solution more straightforward to find.
To solve, we can adopt various methods like substitution, elimination, or matrix approaches such as Cramer's Rule, which will be further discussed.

Determinant Calculation Basics

Before delving into Cramer's Rule, it's crucial to understand how to calculate a "determinant". The determinant is a special number that can be calculated from a square matrix, and it helps in understanding certain properties of the matrix including whether a system has a unique solution.
For a 2x2 matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant is calculated as:

• \( ext{Det}(A) = ad - bc \)

In our exercise:

• Matrix \(A = \begin{pmatrix} 6 & 12 \ 4 & 7 \end{pmatrix}\)
• Determinant: \( ext{Det}(A) = 6 \times 7 - 12 \times 4 = -6 \)

The non-zero determinant tells us that the system of equations has a unique solution, which means one specific set of values for "\(x\)" and "\(y\)" satisfies both equations. The determinant also plays a vital role in the application of Cramer's Rule, which we'll explain next.

Introduction to Matrix Algebra and Cramer's Rule

"Matrix algebra" refers to the use of matrices to perform operations and solve mathematical problems, especially in systems of equations. Cramer's Rule is a method that applies matrix algebra to find solutions by utilizing determinants.
For a given system, you create a matrix of coefficients (like matrix \(A\)), and then modify this matrix to form new ones for each variable in the system. In our example:

• Original matrix \(A = \begin{pmatrix} 6 & 12 \ 4 & 7 \end{pmatrix}\)
• Modified \(A_x\) for \(x\) by replacing the "\(x\)" column with constants: \(\begin{pmatrix} 33 & 12 \ 20 & 7 \end{pmatrix}\)
• Modified \(A_y\) for "\(y\)" by replacing the "\(y\)" column with constants: \(\begin{pmatrix} 6 & 33 \ 4 & 20 \end{pmatrix}\)

Compute the determinant for these matrices:

• \( \text{Det}(A_x) = -9, \quad \text{Det}(A_y) = -12 \)

Finally, Cramer's Rule provides a clear formula for solutions:

• \( x = \frac{\text{Det}(A_x)}{\text{Det}(A)} = 1.5 \)
• \( y = \frac{\text{Det}(A_y)}{\text{Det}(A)} = 2 \)

This rule effectively uses determinants derived from matrix manipulations to solve for variables in a system, showcasing the power and flexibility of matrix algebra in solving equations.