Systems of equations are used in statistics, business, finance, or almost any type of career where two quantities must be compared. Let’s look at an example.
Sara wants to sign up for a telephone service. The cost of a local phone line is $36 per month, plus $0.10 per long distance call. The cost of a cell phone is $50 per month with unlimited long distance, as long as the minutes do not go over 1,000. How many long distance calls would Sara need to make on the local phone line to spend as much as she would if she used a cell phone?
Find the linear equations:
Phone
Monthly
per minute charge
Equation
Telephone
$36
$0.10
y = 0.10x + 36
Cell Phone
$50
no charge
y = 50
We call these two equations together a system of equations. A system of equations is a set of two or more equations for which we want to find a common solution.
One way to find a common solution is to graph the equations and find the common point.
Example 1: Graphing to Find a Common Point
Graph the equations from the telephone example to find the number of long distance minutes that would make the charges for the two phones the same.
y = 36 + 0.10x
y = 50
Using the slope-intercept form of graphing, start with the first equation.
The y-intercept is 36.
The slope is 0.10 = 10/100.
Now, graph y = 50 on the same graph.
The y-intercept is 50.
The slope is zero.
This is a horizontal line.
We can clearly see that they intersect at the point (140, 50), so that is a common solution for both equations.
Therefore, Sara would have to talk for 140 minutes long distance each month for the two charges to come out the same.
Think About It
In the previous example, you saw how to find a common solution by graphing a system of equations. What are the different things that could happen if you graph two lines? How many solutions are possible?
Case 1
The lines could cross each other. Would you ever get more than one point of intersection when they cross if they are straight lines?
There will be exactly one solution point if the two lines intersect.
Case 2
The lines might be parallel. In that case do you get any points of intersection?
There will be no solutions if the two lines are parallel.
Case 3
The two equations might be the equation of the same line, such as:
2x + 2y = 4
and
x + y = 2.
They both are y = -x + 2 when written in slope-intercept form. How many points of intersection do the graphs of the two equations have if they are the same line?
There will be an infinite number of solutions, because the two lines share all points and there are an infinite number of points on any line.
When finding solutions for a system of equations, you may find zero, one or an infinite number of solutions.
If the graphs intersect, the system is said to be consistent.
If the graphs are parallel, the system is said to be inconsistent.
If the system has one solution, it is independent, because the equations are of different lines.
If the system has an infinite number of solutions, the two equations are of the same line and are dependent.
Now we’ll write this as a rule.
A System of Equations Will Have One of the Following Solutions.
Intersecting Lines
One Solution
They are consistent and independent.
Parallel Lines
No Solution
They are inconsistent.
Same Line
Infinite Solutions
They are consistent and dependent.
Example 2: Solving a System
Graph each of the following systems. Determine if there is zero, one, or infinite solutions. If there is one solution, name it.
y = 1/2 x – 2.5
y = -4/3 x + 3
When graphed, there is one solution.
The solution is the point (3, -1).
x + 2y = -1
y = - ½ x + 4
Write the first equation in slope-intercept form to graph it.
x + 2y = -1
2y = -1 – x
y = -½ - ½ x
y = - ½ x + -½
This equation and y = - ½ x + 4 have the same slope, which makes them parallel lines.
There is no solution.
-3x + 3y = -6
6x – 6y = 12
First, write both equations in slope-intercept form, and then graph.
-3x + 3y = -6
3y = -6 + 3x
y = -6/3 + 3/3x
y = -2 + x
y = x – 2
6x – 6y = 12
-6y = 12 – 6x
y = 12/-6 – 6/-6 x
y = -2 + x
y = x – 2
They are both the same equation.
There are infinite solutions.
Think About It
The solution for the system is (3, -1).
y = ½ x – 2.5
y = - 4/3 x + 3.
How can we check the solution, if the solution is one point?
The simplest way is to substitute the values for x and y into each equation and make sure that it does solve both.
y = 1/2 x – 2.5
-1 = 1/2(3) – 2.5
-1 = 1.5 – 2.5
-1 = -1 √
Since both are true statements, the point (3, -1) is the solution for the system.
Need a piece of graph paper?
If so, click here and save it to your computer. That way, you can print off a piece of graph paper whenever you need one. You may find it useful to have this for other parts of the course, as well.
Quick Practice
Is (5, 2) a solution for the system?
2x + 3y = 16
3x – 7y = -1
Think About It
A System of Equations Will Have One of the Following Solutions.
Think About It
