Problem 22 - College Algebra

Question

Use Cramer's Rule to solve each system. $$\left\\{\begin{array}{c}y=-4 x+2 \\\2 x=3 y+8\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is x=0.2 and y=2.8.

Step 1: Formulate the Matrices

Cramer's Rule requires two matrices, A and B. A is the coefficients of the variables (in this case, x and y) and B is the constants on the right side of the equation. The system \[ \left\{ \begin{matrix}y = -4x + 2 \ 2x = 3y+8 \end{matrix} \right. \] rewrites to \[ \left\{ \begin{matrix}-4x + y = 2 \ 2x - 3y = -8 \end{matrix} \right. \]. So, \( A = \left[ \begin{matrix}-4 & 1 \ 2 & -3 \end{matrix}\right] \) and \( B = \left[ \begin{matrix}2 \ -8\end{matrix}\right] \)

Step 2: Calculate the Determinant of A

The determinant of a 2x2 matrix is calculated as \(ad-bc\), where \(a, b, c, d\) are the elements in the matrix. In matrix A, \(a=-4, b=1, c=2, d=-3\). Therefore, \[detA = (-4*-3) - (1*2) = 12 - 2 = 10.\]

Step 3: Calculate the Determinants for x and y

To find the determinants for x and y, replace the first and the second column of A with B, respectively, and calculate the determinant of the resulting matrix. For x, \(Ax = \left[ \begin{matrix}2 & 1 \ -8 & -3\end{matrix}\right]\), and \(detAx = 2*-3 - 1*-8 = -6 - (-8) = 2\). For y, \(Ay = \left[ \begin{matrix}-4 & 2\ 2 & -8\end{matrix}\right]\), and \(detAy = -4*-8 - 2*2 = 32 - 4 = 28.\)

Step 4: Find the solutions for x and y

According to Cramer's rule, the solutions for the variables x and y will be \(x=detAx/detA\) and \(y=detAy/detA\), respectively. Therefore, \(x = 2/10 = 0.2\) and \(y = 28/10 = 2.8.\)

Key Concepts

Understanding Determinants

In the world of linear algebra, a determinant is a special number that we can calculate from a square matrix. It's a powerful tool used to analyze the properties of matrices. This concept is crucial for methods like Cramer's Rule, which is often applied to solve systems of linear equations.

When dealing with a 2x2 matrix, calculating the determinant is straightforward. Suppose we have a matrix \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \], the determinant, denoted as det(A), is calculated by the formula:

• \( ad - bc \)

This simple yet powerful formula helps us find out if a matrix is invertible. If the determinant is not zero, the matrix is invertible, which means we can use it to solve linear equations using Cramer's Rule.

For our system of equations, after forming the matrix A with elements \(a=-4\), \(b=1\), \(c=2\), and \(d=-3\), the determinant comes out to be 10. It confirms that this matrix can indeed be used in Cramer's Rule to find unique solutions for x and y.

The Beauty of a 2x2 Matrix

Matrices are an essential component in mathematics, especially when dealing with linear equations. A 2x2 matrix is a simple form of matrix with two rows and two columns. This type sometimes serves as an excellent entry point for students learning about matrices.

For example, consider a matrix A like this:

• \( A = \begin{bmatrix} -4 & 1 \ 2 & -3 \end{bmatrix} \)

Two main features make a 2x2 matrix special:

• Simplicity: Its small size makes it easier to calculate determinants by hand.
• Practicality: Despite its small size, it is powerful enough to represent basic transformations in a plane.

In our exercise, the matrix A includes coefficients from the linear equations. These simple matrices allow us to compute the determinant and substitute columns to find solutions using techniques like Cramer's Rule.

Systems of Linear Equations Made Simple

Systems of linear equations consist of multiple linear equations involving the same set of variables. They're commonly used in various fields to model relationships between different quantities.

A system like ours, with two equations and two unknowns, can be solved using different methods, one of which is Cramer's Rule. Here is how we set up the problem:

• Equation 1: \( -4x + y = 2 \)
• Equation 2: \( 2x - 3y = -8 \)

This system can be represented by matrices, splitting the system into a coefficient matrix and a constants vector.

To solve such systems, the objective is to find values for x and y that satisfy both equations simultaneously. When using Cramer's Rule, we specifically use determinants to find unique solutions.

The given system indeed has a unique solution since the determinant of the coefficient matrix is non-zero, allowing us to calculate the values of x and y as described. For our equations, these values were found to be \( x = 0.2 \) and \( y = 2.8 \). This offers a straightforward method enabling us to efficiently solve the system by utilizing the power of matrices and determinants.