Problem 21 - Algebra and Trigonometry

Question

Use Cramer’s Rule to solve each system. $$\left\\{\begin{array}{l}{2 x=3 y+2} \\\\{5 x=51-4 y}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution of the system is \(x = \frac{25}{23}\) and \(y = \frac{94}{23}\)

Step 1: Write the system in the form of AX = B

First, the system of equations needs to be rewritten in the matrix form AX = B where A is the matrix of coefficients of the variables, X is the vector of variables, and B is the vector of right-hand side constants. The system can be rewritten as: \[ \left[ \begin{array}{cc} 2 & -3 \ 5 & 4 \ \end{array} \right] \left[ \begin{array}{c} x \ y \ \end{array} \right] = \left[ \begin{array}{c} 2 \ 51 \ \end{array} \right] \]

Step 2: Calculate the determinant of A

The determinant of A is given by the formula \(det(A) = a*d - b*c\), where a, b, c and d are the elements of the matrix A. Hence, \(det(A) = 2*4 - (-3*5) = 23 \)

Step 3: Form matrix D_x by replacing first column of A with B, and calculate its determinant.

The matrix D_x is given by: \[ \begin{array}{cc} 2 & -3 \ 51 & 4 \ \end{array} \] And \(det(D_x) = 2*4 - (-3*51) = 25\)

Step 4: Form matrix D_y by replacing second column of A with B, and calculate its determinant.

The matrix D_y is given by: \[ \begin{array}{cc} 2 & 2 \ 5 & 51 \ \end{array} \] And \(det(D_y) = 2*51 - (2*5) = 94\)

Step 5: Solve for x and y using Cramer's rule

According to Cramer's rule, \(x = \frac{det(D_x)}{det(A)}\) and \(y = \frac{det(D_y)}{det(A)}\). Therefore: \(x = \frac{25}{23} \) and \( y = \frac{94}{23} \)

Key Concepts

Understanding Systems of Equations

A system of equations consists of two or more equations that share the same variables. When we want to solve a system of equations, we are looking for the variable values that satisfy all equations at the same time. In our example, we have:

• Two equations with variables \( x \) and \( y \):
• \( 2x = 3y + 2 \)
• \( 5x = 51 - 4y \)

Systems of equations can represent real-world problems such as finding the point of intersection between two lines on a graph. To solve these systems, various methods can be employed, such as substitution, elimination, and Cramer's Rule. Cramer's Rule is particularly useful for systems with an equal number of equations and variables.

The Role of Determinants in Solving Equations

Determinants are a special number that can be calculated from a square matrix. They offer a lot of insights, especially in linear algebra, to help solve systems of equations. For a 2x2 matrix, the determinant is found using the formula:\[\text{det}(A) = ad - bc\]where \( a, b, c, \) and \( d \) are elements of the matrix.
The determinant tells us how the values in a matrix scale or rotate the space. When using Cramer's Rule, determinants play a key role in finding out whether a solution exists.

• If the determinant of matrix \( A \) is zero, Cramer's Rule cannot be applied as the system doesn't have a unique solution.
• If the determinant is non-zero, the system has a unique solution and we can calculate the variables using matrix equations.

Matrix Algebra Basics

Matrix algebra is essential for handling systems of equations, particularly when dealing with multiple variables. A matrix is essentially a rectangular array of numbers arranged in rows and columns, like a box of elements.
In our case:

• Matrix \( A \) represents the coefficients of our variables.
• Matrix \( X \) is the column matrix of variables \( x \) and \( y \).
• Matrix \( B \) holds the constants from the right side of the equations.

So, the matrix form of our system \( AX = B \) helps us visualize the problem and prepare for solving it with techniques such as Cramer's Rule. Understanding these fundamentals of matrix algebra makes it easier to see how systems of equations can be manipulated to find solutions efficiently.