4: Solving Linear Systems
- Page ID
- 59865
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- 4.1: 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.
- 4.2: 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.
- 4.3: 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.
- 4.4: 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.
- 4.5: 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.