
# 3: Solving Linear Systems

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• 3.1: Linear Systems with Two Variables and Their Solutions
Real-world applications are often modeled using more than one variable and more than one equation. A system of equations consists of a set of two or more equations with the same variables. In this section, we will study linear systems consisting of two linear equations each with two variables.
• 3.2: Solving Linear Systems with Two Variables
In this section, we review a completely algebraic technique for solving systems, the substitution method11. The idea is to solve one equation for one of the variables and substitute the result into the other equation. After performing this substitution step, we are left with a single equation with one variable, which can be solved using algebra.
• 3.3: Applications of Linear Systems with Two Variables
If we translate an application to a mathematical setup using two variables, then we need to form a linear system with two equations. Setting up word problems with two variables often simplifies the entire process, particularly when the relationships between the variables are not so clear.
• 3.4: Solving Linear Systems with Three Variables
We can solve systems of three linear equations with three unknowns by elimination. If the process of solving a system leads to a false statement, then the system is inconsistent and has no solution. If the process of solving a system leads to a true statement, then the system is dependent and has infinitely many solutions.
• 3.5: Matrices and Gaussian Elimination
A linear system in upper triangular form can easily be solved using back substitution. The augmented coefficient matrix and Gaussian elimination can be used to streamline the process of solving linear systems.
• 3.6: Determinants and Cramer’s Rule
A square matrix is a matrix where the number of rows is the same as the number of columns. In this section we outline another method for solving linear systems using special properties of square matrices. Wen introduce the determinant  and show how  Cramer’s rule can be used to efficiently determine solutions to linear systems.
• 3.7: Solving Systems of Inequalities with Two Variables
A system of inequalities consists of a set of two or more inequalities with the same variables. The inequalities define the conditions that are to be considered simultaneously.
• 3.E: Solving Linear Systems