Problem 12 - Algebra 2

Question

Use Cramer’s Rule to solve each system of equations. \(4 x+3 y=6\) \(8 x-y=-9\)

Step-by-Step Solution

Answer

\(x = -\frac{3}{4}, y = 3\)

Step 1: Set Up Coefficient Matrix

First, write down the coefficient matrix of the system of equations. For this system, the matrix is \( A = \begin{bmatrix} 4 & 3 \ 8 & -1 \end{bmatrix} \).

Step 2: Determine Matrix Determinant

Find the determinant of matrix \( A \). The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( ad - bc \). For matrix \( A \), the determinant \( \text{det}(A) = 4(-1) - 3(8) = -4 - 24 = -28 \).

Step 3: Create Matrix for x

To find \( x \) using Cramer's Rule, replace the first column of matrix \( A \) with the constants from the right-hand side of the equations. This gives \( A_x = \begin{bmatrix} 6 & 3 \ -9 & -1 \end{bmatrix} \).

Step 4: Determine x-Matrix Determinant

Find the determinant of \( A_x \). Use the formula \( ad - bc \) for this matrix. So, \( \text{det}(A_x) = 6(-1) - 3(-9) = -6 + 27 = 21 \).

Step 5: Calculate x using Cramer's Rule

Using Cramer's Rule, \( x = \frac{\text{det}(A_x)}{\text{det}(A)} \). Thus, \( x = \frac{21}{-28} = -\frac{3}{4} \).

Step 6: Create Matrix for y

Replace the second column of matrix \( A \) with the constants from the right-hand side to find \( y \). This gives \( A_y = \begin{bmatrix} 4 & 6 \ 8 & -9 \end{bmatrix} \).

Step 7: Determine y-Matrix Determinant

Find the determinant of \( A_y \). So, \( \text{det}(A_y) = 4(-9) - 6(8) = -36 - 48 = -84 \).

Step 8: Calculate y using Cramer's Rule

Using Cramer's Rule, \( y = \frac{\text{det}(A_y)}{\text{det}(A)} \). Thus, \( y = \frac{-84}{-28} = 3 \).

Key Concepts

System of Equations

A system of equations consists of two or more equations with multiple variables that need to be solved simultaneously. In this exercise, we have a system with two equations:

• Equation 1: \(4x + 3y = 6\)
• Equation 2: \(8x - y = -9\)

These equations intersect at a specific point in a coordinate system. To solve them, we seek a common value for the variables \(x\) and \(y\) that satisfies both equations. The point of intersection represents the solution to the system. Some methods to solve systems of equations are substitution, elimination, and as shown in this exercise, Cramer's Rule. When using Cramer's Rule, it's essential to verify that the determinant of the coefficient matrix is non-zero to ensure a unique solution exists.

Determinant

The determinant is a special number calculated from a square matrix. It is denoted as \( ext{det}(A) \) for a matrix \( A \). The determinant plays a crucial role in various matrix operations, including solving systems of linear equations as we've seen with Cramer's Rule.
The determinant can provide information on the solvability of the system:

• If the determinant is zero, the system may be either inconsistent or have infinitely many solutions.
• If it is non-zero, there is a unique solution.

In the exercise, we calculate the determinant of the coefficient matrix \( A = \begin{bmatrix} 4 & 3 \ 8 & -1 \end{bmatrix} \). Using the formula for a 2x2 matrix, the determinant is \( ad - bc = 4(-1) - 3(8) = -28 \). Since it is not zero, Cramer's Rule can be applied successfully to find the unique solutions for \( x \) and \( y \).

Matrix Operations

Matrix operations are fundamental tools in linear algebra, allowing us to perform calculations on systems of equations. In the current exercise, we primarily deal with the formation of matrices and the calculation of their determinants, which are core matrix operations.
Here is a brief overview of relevant matrix operations:

• Matrix Formation: Writing down rows and columns of numbers, known as elements, to form a matrix.
• Determinant Calculation: Specifically for 2x2 matrices, applying the formula \( ad - bc \) to find the determinant.
• Matrix Column Replacement: In Cramer's Rule, replacing columns of the coefficient matrix with the constant columns to create new matrices like \( A_x \) and \( A_y \).
• Solving Matrices: Using determinants to find solutions for the variables in the equations using Cramer's Rule.

Making and manipulating these matrices effectively allows us to solve complex systems of equations efficiently.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and linear equations. It provides the theoretical framework for solving systems of equations, which is essential in various fields, from computer science to engineering.
In the context of this exercise, linear algebra principles such as matrices and determinants are key concepts. Here’s how they connect:

• Matrices: Represent linear transformations and systems of equations in a compact form.
• Determinants: Provide valuable insight into the nature of these systems, such as whether they have unique solutions.
• Cramer's Rule: An application of linear algebra that employs determinants and matrix operations to solve systems of linear equations.
• Vector Spaces: Though not directly covered here, understanding that linear combinations of row or column vectors in matrices form vector spaces is crucial.

By harnessing these concepts, we can solve practical problems involving multiple unknowns and linear relationships, as illustrated in this exercise.