Problem 43 - Algebra 2

Question

Use Cramer’s Rule to solve each system of equations. \(6 x+7 y=10\) \(3 x-4 y=20\)

Step-by-Step Solution

Answer

The solution is \(x = 4\) and \(y = -2\).

Step 1: Write the equations in standard form

The system of equations is already given in standard form: \(6x + 7y = 10\) and \(3x - 4y = 20\).

Step 2: Determine the coefficient matrix

Identify the coefficients of the variables from the system of equations. The coefficient matrix \(A\) is: \[A = \begin{pmatrix} 6 & 7 \ 3 & -4 \end{pmatrix}\]

Step 3: Calculate the determinant of A

To use Cramer's Rule, compute the determinant of the matrix \(A\):\[\text{det}(A) = \begin{vmatrix} 6 & 7 \ 3 & -4 \end{vmatrix} = (6 \cdot -4) - (3 \cdot 7) = -24 - 21 = -45\]

Step 4: Form the matrices for x and y

Replace the relevant columns with the constants from the equations:\[A_x = \begin{pmatrix} 10 & 7 \ 20 & -4 \end{pmatrix}, \; A_y = \begin{pmatrix} 6 & 10 \ 3 & 20 \end{pmatrix}\]

Step 5: Calculate the determinant of A_x

Find the determinant of \(A_x\):\[\text{det}(A_x) = \begin{vmatrix} 10 & 7 \ 20 & -4 \end{vmatrix} = (10 \cdot -4) - (20 \cdot 7) = -40 - 140 = -180\]

Step 6: Calculate the determinant of A_y

Find the determinant of \(A_y\):\[\text{det}(A_y) = \begin{vmatrix} 6 & 10 \ 3 & 20 \end{vmatrix} = (6 \cdot 20) - (3 \cdot 10) = 120 - 30 = 90\]

Step 7: Solve for x using Cramer's Rule

Using Cramer's Rule, the solution for \(x\) is:\[x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{-180}{-45} = 4\]

Step 8: Solve for y using Cramer's Rule

Using Cramer's Rule, the solution for \(y\) is:\[y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{90}{-45} = -2\]

Step 9: State the solution

The solution to the system of equations is \(x = 4\) and \(y = -2\).

Key Concepts

System of Equations

A system of equations involves a set of two or more equations that share common variables and need to be solved together. In our exercise, the system comprises two equations: \(6x + 7y = 10\) and \(3x - 4y = 20\). The goal is to find values for \(x\) and \(y\) that satisfy both equations simultaneously. Such systems can involve linear or non-linear equations, and can have one solution, infinitely many solutions, or no solution at all. In this case, with linear equations, it's crucial to express each equation in standard form, \(Ax + By = C\), to apply solving techniques like Cramer's Rule effectively.

• Each equation represents a line in a two-dimensional plane.
• Their intersection point(s) - if any - provides the solutions for \(x\) and \(y\).
• Techniques like Cramer's Rule or substitution are used to find the solution.

Understanding this concept is vital for solving systems of equations successfully.

Determinant

To solve a system of equations using Cramer's Rule, calculating the determinant of the coefficient matrix is a fundamental step. The determinant is a scalar value that can be computed from the elements of a square matrix. It plays a key role in understanding whether a system of equations has a unique solution: if the determinant is zero, the system may not have a unique solution. In our exercise, the coefficient matrix \(A\) is \(\begin{pmatrix} 6 & 7 \ 3 & -4 \end{pmatrix}\), and its determinant is calculated as follows:

\[\text{det}(A) = \begin{vmatrix} 6 & 7 \ 3 & -4 \end{vmatrix} = (6 \cdot -4) - (3 \cdot 7) = -24 - 21 = -45\]

Key points about determinants include:

• A non-zero determinant indicates a unique solution exists.
• Determinants can also help find inverse matrices and solve systems of equations.
• They are only defined for square matrices (same number of rows and columns).

This concept is essential for using Cramer's Rule.

Coefficient Matrix

The coefficient matrix is derived from the system of equations and is crucial for solving systems using Cramer's Rule or other matrix-based methods. It is constructed by taking the coefficients of the variables from each equation and arranging them in matrix form. For the given system \(6x + 7y = 10\) and \(3x - 4y = 20\), the coefficient matrix \(A\) looks like this:

\[A = \begin{pmatrix} 6 & 7 \ 3 & -4 \end{pmatrix}\]

Understanding the coefficient matrix includes:

• It organizes coefficients into a structured format that aids computational methods.
• Only coefficients of the variables are included, not the constants.
• The matrix must be square for applications like Cramer's Rule.

This matrix serves as the cornerstone for calculating determinants and further solving the system using matrix algebra.

Matrix Algebra

Matrix algebra provides a powerful framework for solving systems of linear equations, among other applications. It involves operations like addition, subtraction, and multiplication on matrices, which can represent mathematical structures concisely. In our case, after formulating the coefficient matrix \(A\), matrix algebra allows us to use Cramer's Rule efficiently to find the solutions for the system. Using matrices, we can:

• Compute determinants, which help determine solutions' existence and uniqueness.
• Replace columns to compute intermediate matrices needed for Cramer's Rule.
• Use inverse matrices, if applicable, to solve equations by multiplication.

For instance, we find determinants \(\text{det}(A_x)\) and \(\text{det}(A_y)\) after constructing specific matrices by replacing columns. Understanding matrix algebra is crucial because it extends mathematical capabilities beyond numbers, allowing systematic approaches to complex problems.