# Systems of Linear Equations

## Graphing Systems of Linear Inequalities

 

### Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.
A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown below.
To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs  that make both inequalities true.
To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.

### Solve a System of Linear Inequalities by Graphing

The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.
Systems of linear inequalities where the boundary lines are parallel might have no solution. We’ll see this in .

### Solve Applications of Systems of Inequalities

The first thing we’ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system as we did above to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so their graphs will only show Quadrant I.

### Key Concepts

1. To Solve a System of Linear Inequalities by Graphing



### Section Exercises


### Practice Makes Perfect

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities
In the following exercises, determine whether each ordered pair is a solution to the system.
Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.
Solve Applications of Systems of Inequalities
In the following exercises, translate to a system of inequalities and solve.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

### Chapter 5 Review Exercises


### Solve Systems of Equations by Graphing

Determine Whether an Ordered Pair is a Solution of a System of Equations.
In the following exercises, determine if the following points are solutions to the given system of equations.
Solve a System of Linear Equations by Graphing
In the following exercises, solve the following systems of equations by graphing.
Determine the Number of Solutions of a Linear System
In the following exercises, without graphing determine the number of solutions and then classify the system of equations.
Solve Applications of Systems of Equations by Graphing

### Solve Systems of Equations by Substitution

Solve a System of Equations by Substitution
In the following exercises, solve the systems of equations by substitution.
Solve Applications of Systems of Equations by Substitution
In the following exercises, translate to a system of equations and solve.

### Solve Systems of Equations by Elimination

Solve a System of Equations by Elimination
In the following exercises, solve the systems of equations by elimination.
Solve Applications of Systems of Equations by Elimination
In the following exercises, translate to a system of equations and solve.
Choose the Most Convenient Method to Solve a System of Linear Equations
In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

### Solve Applications with Systems of Equations

Translate to a System of Equations
In the following exercises, translate to a system of equations. Do not solve the system.
Solve Direct Translation Applications
In the following exercises, translate to a system of equations and solve.
Solve Geometry Applications
In the following exercises, translate to a system of equations and solve.
Solve Uniform Motion Applications
In the following exercises, translate to a system of equations and solve.

### Solve Mixture Applications with Systems of Equations

Solve Mixture Applications
In the following exercises, translate to a system of equations and solve.
Solve Interest Applications
In the following exercises, translate to a system of equations and solve.

### Graphing Systems of Linear Inequalities

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities
In the following exercises, determine whether each ordered pair is a solution to the system.
Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.
Solve Applications of Systems of Inequalities
In the following exercises, translate to a system of inequalities and solve.

### Practice Test

In the following exercises, solve the following systems by graphing.
In the following exercises, solve each system of equations. Use either substitution or elimination.
In the following exercises, translate to a system of equations and solve.