For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases}x + y = 7 \\x - y = 3\end{cases}$

Verified step by step guidance

1. Write the system of equations in standard form: \\ \(x + y = 7\) and \(x - y = 3\).
2. Identify the coefficient matrix \(A\), the variable matrix \(X\), and the constant matrix \(B\): \\ \(A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\), \(X = \begin{bmatrix} x \\ y \end{bmatrix}\), \(B = \begin{bmatrix} 7 \\ 3 \end{bmatrix}\).
3. Calculate the determinant of the coefficient matrix \(A\): \\ \(D = \det(A) = (1)(-1) - (1)(1)\).
4. Form matrices \(A_x\) and \(A_y\) by replacing the respective columns of \(A\) with the constant matrix \(B\): \\ \(A_x = \begin{bmatrix} 7 & 1 \\ 3 & -1 \end{bmatrix}\) and \(A_y = \begin{bmatrix} 1 & 7 \\ 1 & 3 \end{bmatrix}\).
5. Calculate the determinants \(D_x = \det(A_x)\) and \(D_y = \det(A_y)\), then use Cramer's Rule to find the variables: \\ \(x = \frac{D_x}{D}\) and \(y = \frac{D_y}{D}\).

Key Concepts

Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It applies to square systems where the number of equations equals the number of variables. The solution for each variable is found by replacing the corresponding column of the coefficient matrix with the constants vector and dividing the determinant of this new matrix by the determinant of the coefficient matrix.

Determinants of 2x2 Matrices

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This scalar value helps determine if the system has a unique solution (non-zero determinant) or not. In Cramer's Rule, determinants are used to find the values of variables by comparing the determinant of the coefficient matrix and modified matrices.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. Understanding how to represent and manipulate these systems is essential for applying methods like Cramer's Rule.