# Systems of Linear Equations

## Solve Systems of Equations by Graphing


### Determine Whether an Ordered Pair is a Solution of a System of Equations

In Solving Linear Equations and Inequalities we learned how to solve linear equations with one variable. Remember that the solution of an equation is a value of the variable that makes a true statement when substituted into the equation.
Now we will work with systems of linear equations, two or more linear equations grouped together.
We will focus our work here on systems of two linear equations in two unknowns. Later, you may solve larger systems of equations.
An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.
A linear equation in two variables, like 2x + y = 7, has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.
To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs (x, y) that make both equations true. These are called the solutions to a system of equations.
To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.
Let’s consider the system below:
Is the ordered pair  a solution?
The ordered pair (2, −1) made both equations true. Therefore (2, −1) is a solution to this system.
Let’s try another ordered pair. Is the ordered pair (3, 2) a solution?
The ordered pair (3, 2) made one equation true, but it made the other equation false. Since it is not a solution to both equations, it is not a solution to this system.

### Solve a System of Linear Equations by Graphing

In this chapter we will use three methods to solve a system of linear equations. The first method we’ll use is graphing.
The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we’ll find the solution to the system.
Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions.
Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown in :
For the first example of solving a system of linear equations in this section and in the next two sections, we will solve the same system of two linear equations. But we’ll use a different method in each section. After seeing the third method, you’ll decide which method was the most convenient way to solve this system.
The steps to use to solve a system of linear equations by graphing are shown below.
Both equations in  were given in slope–intercept form. This made it easy for us to quickly graph the lines. In the next example, we’ll first re-write the equations into slope–intercept form.
Usually when equations are given in standard form, the most convenient way to graph them is by using the intercepts. We’ll do this in .
Do you remember how to graph a linear equation with just one variable? It will be either a vertical or a horizontal line.
In all the systems of linear equations so far, the lines intersected and the solution was one point. In the next two examples, we’ll look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.
If you write the second equation in  in slope-intercept form, you may recognize that the equations have the same slope and same y-intercept.
When we graphed the second line in the last example, we drew it right over the first line. We say the two lines are coincident. Coincident lines have the same slope and same y-intercept.
 

### Determine the Number of Solutions of a Linear System

There will be times when we will want to know how many solutions there will be to a system of linear equations, but we might not actually have to find the solution. It will be helpful to determine this without graphing.
We have seen that two lines in the same plane must either intersect or are parallel. The systems of equations in  through  all had two intersecting lines. Each system had one solution.
A system with parallel lines, like , has no solution. What happened in ? The equations have coincident lines, and so the system had infinitely many solutions.
We’ll organize these results in  below:
Parallel lines have the same slope but different y-intercepts. So, if we write both equations in a system of linear equations in slope–intercept form, we can see how many solutions there will be without graphing! Look at the system we solved in .
The two lines have the same slope but different y-intercepts. They are parallel lines.
 shows how to determine the number of solutions of a linear system by looking at the slopes and intercepts.
Let’s take one more look at our equations in  that gave us parallel lines.
When both lines were in slope-intercept form we had:
Do you recognize that it is impossible to have a single ordered pair  that is a solution to both of those equations?
We call a system of equations like this an inconsistent system. It has no solution.
A system of equations that has at least one solution is called a consistent system.
We also categorize the equations in a system of equations by calling the equations independent or dependent. If two equations are independent equations, they each have their own set of solutions. Intersecting lines and parallel lines are independent.
If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations, we get coincident lines.
Let’s sum this up by looking at the graphs of the three types of systems. See  and .

### Solve Applications of Systems of Equations by Graphing

We will use the same problem solving strategy we used in Math Models to set up and solve applications of systems of linear equations. We’ll modify the strategy slightly here to make it appropriate for systems of equations.

Step 5 is where we will use the method introduced in this section. We will graph the equations and find the solution.

### Key Concepts

1. To solve a system of linear equations by graphing


2. Determine the number of solutions from the graph of a linear system



3. Determine the number of solutions of a linear system by looking at the slopes and intercepts



4. Determine the number of solutions and how to classify a system of equations



5. Problem Solving Strategy for Systems of Linear Equations



### Practice Makes Perfect

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 Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.
Solve Applications of Systems of Equations by Graphing In the following exercises, solve.

### Everyday Math


### Writing Exercises


### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.