Problem 31 - College Algebra

Question

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} 4 x-3 y=-3 \\ 2 x+6 y=-4 \end{array} $$

Step-by-Step Solution

Answer

The solution is \(x = -1\) and \(y = -\frac{1}{3}\).

Step 1: Write the System in Matrix Form

First, express the system of linear equations in the form of a matrix equation. The system \(4x - 3y = -3\) and \(2x + 6y = -4\) can be written as: \[\begin{bmatrix}4 & -3 \2 & 6\end{bmatrix}\begin{bmatrix}x \y\end{bmatrix}=\begin{bmatrix}-3 \-4\end{bmatrix}\] where \(A = \begin{bmatrix} 4 & -3 \ 2 & 6 \end{bmatrix}\), \(\mathbf{x} = \begin{bmatrix} x \ y \end{bmatrix}\), and \(\mathbf{b} = \begin{bmatrix} -3 \ -4 \end{bmatrix}\).

Step 2: Calculate Determinant of Matrix A

Use the determinant formula \(\text{det}(A) = a_{11}a_{22} - a_{12}a_{21}\) for the matrix \(A\). For our matrix \(A = \begin{bmatrix} 4 & -3 \ 2 & 6 \end{bmatrix}\), calculate \(\text{det}(A) = (4)(6) - (-3)(2) = 24 + 6 = 30\).

Step 3: Calculate Determinants of Matrices for x and y

To find \(x\), replace the first column of \(A\) with \(\mathbf{b}\) forming matrix \(A_x\), and find its determinant. Similarly, for \(y\), replace the second column of \(A\) with \(\mathbf{b}\) to form \(A_y\). Matrix \(A_x = \begin{bmatrix} -3 & -3 \ -4 & 6 \end{bmatrix}\) with \(\text{det}(A_x) = (-3)(6) - (-3)(-4) = -18 + 12 = -30\).Matrix \(A_y = \begin{bmatrix} 4 & -3 \ 2 & -4 \end{bmatrix}\) with \(\text{det}(A_y) = (4)(-4) - (-3)(2) = -16 + 6 = -10\).

Step 4: Apply Cramer's Rule to Find x and y

According to Cramer's Rule, \(x = \frac{\text{det}(A_x)}{\text{det}(A)}\) and \(y = \frac{\text{det}(A_y)}{\text{det}(A)}\).Calculate \(x = \frac{-30}{30} = -1\) and \(y = \frac{-10}{30} = -\frac{1}{3}\).

Step 5: Confirm the Solution

Substitute the solutions \(x = -1\) and \(y = -\frac{1}{3}\) back into the original equations to ensure they satisfy both. For the first equation: \(4(-1) - 3(-\frac{1}{3}) = -4 + 1 = -3\).For the second equation: \(2(-1) + 6(-\frac{1}{3}) = -2 - 2 = -4\). Both equations are satisfied, confirming the solution is correct.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. In this context, we focus on equations with two variables, such as the ones provided in the exercise: \(4x - 3y = -3\) and \(2x + 6y = -4\). These equations, when solved together, give values for \(x\) and \(y\) that satisfy both equations simultaneously.

Solving such systems is fundamental in various fields, including mathematics, physics, engineering, and economics. There are several methods to solve them, with Cramer's Rule being one of them, especially when the system has the same number of equations as unknowns.

Determinant

The determinant is a special number calculated from a square matrix. It gives important properties of the matrix, such as whether it is invertible and information about the geometry of the system represented by the matrix.

For a 2x2 matrix \(A\) represented as \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant \(\text{det}(A)\) is calculated as \(ad - bc\).

• If \(\text{det}(A) = 0\), the matrix is singular, meaning the system of equations has no unique solutions.
• If \(\text{det}(A) eq 0\), the system has a unique solution, and the matrix can be inverted.

In our specific problem, the determinant of matrix \(A\), \(\begin{bmatrix} 4 & -3 \ 2 & 6 \end{bmatrix}\), is found to be \(30\), indicating a unique solution exists for the system.

Matrix Form

Expressing a system of equations in matrix form involves writing the equations as a matrix equation, facilitating easier manipulation and solution.

For our system, the equations \(4x - 3y = -3\) and \(2x + 6y = -4\) can be rewritten in matrix form as:

• Matrix \(A\), representing coefficients of variables, is \(\begin{bmatrix} 4 & -3 \ 2 & 6 \end{bmatrix}\).
• Column vector \(\mathbf{x}\), representing the unknowns, is \(\begin{bmatrix} x \ y \end{bmatrix}\).
• Column vector \(\mathbf{b}\), representing constant terms, is \(\begin{bmatrix} -3 \ -4 \end{bmatrix}\).

The matrix equation takes the form \(A\mathbf{x} = \mathbf{b}\). Rewriting in this form makes it easier to apply linear algebra techniques such as Cramer's Rule for solving.

Matrix Equation

A matrix equation is a compact and efficient way to represent a system of linear equations using matrices. The general form of a matrix equation is \(A\mathbf{x} = \mathbf{b}\), where:

• \(A\) is the coefficient matrix.
• \(\mathbf{x}\) is the column vector of variables.
• \(\mathbf{b}\) is the column vector of constants.

Solving a matrix equation can be done through various methods, with Cramer's Rule being effective when finding individual solutions for each variable. The process involves calculating determinants for the original matrix and matrices modified by replacing certain columns with the constant vector \(\mathbf{b}\).

Understanding matrix equations is vital as they provide a clear structure for solving complex systems, underpin many areas of science and engineering, and are fundamental in computational problem-solving approaches.