Systems of Equations and Inequalities

Solving Systems of Linear Equations

Systems of equations are sets of two or more equations with the same variables. The solution to a system is the set of variable values that satisfy all equations simultaneously. There are several methods to solve such systems, including substitution, elimination, and matrix methods such as Cramer's Rule.

• Cramer's Rule: A method that uses determinants to solve a system of linear equations, applicable when the system has the same number of equations as unknowns and the determinant of the coefficient matrix is nonzero.

• Substitution Method: Involves solving one equation for one variable and substituting this expression into the other equation(s).

Example System

Consider the following system of equations:

Solving with Cramer's Rule

For a system:

We define the following determinants:

The solutions are:

For the given system:

• , ,

• , ,

Calculate the determinants:

Thus, the solution is:

Alternative Methods

• If Cramer's Rule does not apply (e.g., determinant is zero), use substitution or elimination.

• Substitution: Solve one equation for one variable, substitute into the other, and solve for the remaining variable.

Key Points

• Systems can have one solution (consistent and independent), no solution (inconsistent), or infinitely many solutions (dependent).

• Cramer's Rule is only applicable when the determinant of the coefficient matrix is nonzero.

Table: Types of Solutions for Systems of Equations

Additional info: The original file is a review exam question focusing on solving systems of equations using Cramer's Rule and substitution, which is a core topic in College Algebra (Ch. 5).