Problem 39 - Intermediate Algebra

Question

Use Cramer's rule to solve each system of equations. $$ \left\\{\begin{array}{l} x+y=6 \\ x-y=2 \end{array}\right. $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Coefficients Matrix

To use Cramer's rule, first identify the coefficients of the variables from the system of equations. The given system is \( \begin{cases} x + y = 6 \ x - y = 2 \end{cases} \). The coefficients for \( x \) and \( y \) can be represented as a matrix: \[A = \begin{bmatrix} 1 & 1 \ 1 & -1 \end{bmatrix}\]

Step 2: Find the Determinant of the Coefficients Matrix

The determinant of matrix \( A \), denoted \( \det(A) \), is calculated as follows: \[\det(A) = (1)(-1) - (1)(1) = -1 - 1 = -2\]

Step 3: Form the Constants Matrix

The constants from the right-hand side of the equations form the matrix:\[B = \begin{bmatrix} 6 \ 2 \end{bmatrix}\]

Step 4: Compute Determinant for X Matrix

To find the determinant for \( x \), replace the first column of \( A \) with \( B \) to form matrix \( A_x \):\[A_x = \begin{bmatrix} 6 & 1 \ 2 & -1 \end{bmatrix}\]Calculate \( \det(A_x) \):\[\det(A_x) = (6)(-1) - (2)(1) = -6 - 2 = -8\]

Step 5: Compute Determinant for Y Matrix

For \( y \), replace the second column of \( A \) with \( B \) to form matrix \( A_y \):\[A_y = \begin{bmatrix} 1 & 6 \ 1 & 2 \end{bmatrix}\]Calculate \( \det(A_y) \):\[\det(A_y) = (1)(2) - (1)(6) = 2 - 6 = -4\]

Step 6: Solve for X and Y Using Cramer's Rule

Using Cramer's Rule, find \( x \) and \( y \):\[x = \frac{\det(A_x)}{\det(A)} = \frac{-8}{-2} = 4\]\[y = \frac{\det(A_y)}{\det(A)} = \frac{-4}{-2} = 2\]

Step 7: Solution Confirmation

Finally, confirm the solution by substituting \( x = 4 \) and \( y = 2 \) back into the original equations:1. \( 4 + 2 = 6 \) is correct.2. \( 4 - 2 = 2 \) is correct.The solution \( x = 4 \), \( y = 2 \) satisfies both equations.

Key Concepts

Understanding Systems of Equations

In math, systems of equations are collections of two or more equations with a set of variables. They represent several conditions that some variables should satisfy at the same time. You'll typically encounter two types:

• Linear systems: Equations involve first-degree variables. They graph as straight lines when plotted.
• Non-linear systems: These include equations with polynomial elements, powers, or other non-linear features.

Approaching these systems involves finding one solution set that satisfies all the conditions. In the example, we solved a linear system with two equations: \(x + y = 6\) and \(x - y = 2\). By solving this system, you identify values for \(x\) and \(y\) that work in both equations.

Determinant Calculation Essentials

Determinants are a fundamental concept in linear algebra, associated with square matrices. They are a scalar value that provides useful information about a matrix, such as whether the matrix is invertible. When dealing with a 2x2 matrix like \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant, denoted \(\det(A)\), is determined by the formula: \[\det(A) = ad - bc.\]This calculation can help you solve systems of equations using matrix methods like Cramer's Rule. In our problem, we found the determinant of matrix \(A\) as \(-2\) using the given coefficients of \(x\) and \(y\). Understanding how to compute the determinant is critical for identifying system consistency and solutions.

Matrix Methods with Cramer's Rule

Matrix methods provide a structured and efficient way to solve systems of equations. Cramer's Rule is one of these techniques and it relies heavily on determinants. Here's the basic idea:

• Identify each equation's coefficients and form the coefficients matrix \(A\).
• Calculate the determinant of \(A\), which tells us if a unique solution exists.
• Form matrices by replacing columns with the constant terms from your equations. This gives you new matrices for calculating specific variable values. For instance, we created \(A_x\) and \(A_y\) by swapping columns with constants \(B\).

By applying determinants to these new matrices, we isolated the values of each variable. Cramer's Rule is particularly helpful when you have a small number of equations and need exact solutions quickly.

Exploring Algebraic Techniques

Algebraic techniques form the backbone of solving equations. When solving systems of equations, fundamental algebraic manipulation plays a crucial role. Consider where algebra fits into Cramer's Rule:

• Formulating equations and matrices requires precise algebraic understanding.
• Swapping out specific columns in matrices involves careful arithmetic operation.
• Finally, calculating determinants and solving resulting ratios is rooted deeply in algebra.

In our exercise, replacing columns to find \(A_x\) and \(A_y\) requires careful operations to ensure no missteps. Such techniques are vital for ensuring the calculations follow the logical sequence of operations integral to successfully solving the system of equations.