4: Systems of Linear Equations in Two and Three Variables

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Learning Objectives

By the end of this chapter, the student should be able to

Solve a system of equations with two and three linear equations in two and three variables by graphing, substitution, and elimination including infinitely many solutions or no solution
Solve applications involving systems of equations including mixture, value, distance, and interest problems
Graph and find the solutions for systems of two linear inequalities in two variables
Use matrices to solve systems of two linear equations in two variables

We have solved linear equations like \(3x − 4 = 11\) by adding \(4\) to both sides and then dividing by \(3\) (solution is \(x = 5\)). Notice, we only have one variable in this equation. What if we have \(2\) variables? Luckily, we have methods to solve equations with more than one variable. It turns out that to solve for more than one variable we will need the same number of equations as variables. For example, to solve for two variables, such as \(x\) and \(y\), we will need two equations with the same variables. When solving for more than one equation and one variable, we call the set of equations a system of equations. When solving a system of equations, we are looking for a solution that makes both equations true. Since we are solving for \(x\) and \(y\), it should remind us of graphing lines, and the solution is an ordered pair \((x, y)\). This ordered-pair is on both lines.

Definition: System of Two Linear Equations in Two Variables

A system of two linear equations in two variables is given in the form \[\left\{\begin{array}{l}ax+by=c \\ dx+ey=f\end{array}\right.\nonumber\] where \(a,\: b,\: c,\: d,\: e,\) and \(f\) are coefficients and \(x\) and \(y\) are variables. This system is represented in standard form.

4.1: System of Equations - Graphing
4.2: Systems of Equations - The Substitution Method
Solving a system by graphing has its limitations. We rarely use graphing to solve systems. Instead, we use an algebraic approach. There are two approaches and the first approach is called substitution. We build the concepts of substitution through several examples and then conclude with a general four-step process to solve problems using this method.
4.3: System of Equations - The Addition Method
The substitution method is often used for solving systems in various areas of algebra. However, substitution can get quite involved, especially if there are fractions because this only allows more room for error. Hence, we need an even more sophisticated way for solving systems in general. We call this method the addition method, also called the elimination method. We will build the concept in the following examples, then define a four-step process we can use to solve by elimination.
4.4: Applications with systems of equations
We saw these types of examples in a previous chapter, but with one variable. In this section, we review the same types of applications, but solving in a more sophisticated way using systems of equations. Once we set up the system, we can solve using any method we choose. However, setting up the system may be the challenge, but as long as we follow the method we used before, we will be fine. We use tables to organize the parameters.
4.5: Systems of three linear equations in three variables
Solving systems of linear equations in three variables is very similar to the methods in which we solve linear systems in two variables. With linear systems in two variables, we reduced the system down to one linear equation in one variable. With linear systems in three variables, we apply the same method except we reduce the system down from three linear equations in three variables to two linear equations in two variables first, then to one linear equation in one variable.
4.6: Systems of two linear inequalities in two variables
In a previous section, we discussed linear inequalities in two variables, where we have the boundary line, dashed or solid, and shading either above or below the y -intercept, depending on the inequality symbol. Well, let’s use this same idea for finding the solution to a system of two linear inequalities in two variables.
4.7: Systems of Equations- Answers to the Homework Exercises