In Exercises 52–55, use Cramer's Rule to solve each system.

Verified step by step guidance

1. Identify the system of linear equations you need to solve. Let's assume the system is: $ax+by=e$ and $cx+dy=f$.
2. Write the coefficient matrix $A$ for the system, which is $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$.
3. Calculate the determinant of matrix $A$, denoted as $\text{det}(A)=ad-bc$.
4. Create the matrices $A_x$ and $A_y$ by replacing the respective columns of $A$ with the constants from the right-hand side of the equations. $A_x=\left[\begin{array}{cc}e& b\\ f& d\end{array}\right]$ and $A_y=\left[\begin{array}{cc}a& e\\ c& f\end{array}\right]$.
5. Calculate the determinants $\text{det}(A_x)$ and $\text{det}(A_y)$, then use Cramer's Rule to find $x=\frac{\text{det}(A_x)}{\text{det}(A)}$ and $y=\frac{\text{det}(A_y)}{\text{det}(A)}$.

Key Concepts

Cramer's Rule

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, provided the determinant of the coefficient matrix is non-zero. It expresses the solution of the system in terms of determinants, allowing for the calculation of each variable by substituting the constant terms into the determinant of the coefficient matrix.

Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix and provides important properties of the matrix, such as whether it is invertible. In the context of Cramer's Rule, the determinant of the coefficient matrix is crucial for determining if a unique solution exists for the system of equations.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The solution to such a system is the set of values for the variables that satisfy all equations simultaneously. Understanding how to represent and manipulate these systems is essential for applying Cramer's Rule effectively.