# 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 here.
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 the next example.
Some systems of linear inequalities where the boundary lines are parallel will have a solution. We’ll see this in the next example.

### 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 we add inequalities to the system as additional requirements.
When we use variables other than x and y to define an unknown quantity, we must change the names of the axes of the graph as well.

### Key Concepts

1. Solutions of a System of Linear Inequalities: Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true. The solution of a system of linear inequalities is shown as a shaded region in the x, y coordinate system that includes all the points whose ordered pairs make the inequalities true.
2. How 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.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

### Chapter Review Exercises


### Solve Systems of Linear Equations with Two Variables

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.
In the following exercises, without graphing determine the number of solutions and then classify the system of equations.
Solve a System of Equations by Substitution
In the following exercises, solve the systems of equations by substitution.
Solve a System of Equations by Elimination
In the following exercises, solve the systems of equations by elimination
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

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 with Systems of Equations
For the following exercises, translate to a system of equations and solve.
Solve Interest Applications
For the following exercises, translate to a system of equations and solve.

### Solve Systems of Equations with Three Variables

Solve Systems of Equations with Three Variables
In the following exercises, determine whether the ordered triple is a solution to the system.
Solve a System of Linear Equations with Three Variables
In the following exercises, solve the system of equations.
Solve Applications using Systems of Linear Equations with Three Variables

### Solve Systems of Equations Using Matrices

Write the Augmented Matrix for a System of Equations.
Write each system of linear equations as an augmented matrix.
Write the system of equations that that corresponds to the augmented matrix.
In the following exercises, perform the indicated operations on the augmented matrices.
Solve Systems of Equations Using Matrices
In the following exercises, solve each system of equations using a matrix.

### Solve Systems of Equations Using Determinants

Evaluate the Determinant of a 2 × 2 Matrix
In the following exercise, evaluate the determinate of the square matrix.
Evaluate the Determinant of a 3 × 3 Matrix
In the following exercise, find and then evaluate the indicated minors.
In the following exercise, evaluate each determinant by expanding by minors along the first row.
In the following exercise, evaluate each determinant by expanding by minors.
Use Cramer’s Rule to Solve Systems of Equations
In the following exercises, solve each system of equations using Cramer’s rule
Solve Applications Using Determinants
In the following exercises, determine whether the given points are collinear.

### 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.

### Chapter 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.
Solve the system of equations using a matrix.
Solve using Cramer’s rule.
In the following exercises, translate to a system of equations and solve.