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In this lesson, it will be shown how a graph can help find values of variables that satisfy two linear equations simultaneously.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Here are a few practice exercises before getting started with this lesson.

a Write the equation in slope-intercept form.
b Consider the following linear graphs.
linear graphs
linear graphs
Which graph corresponds to
Challenge

Basketball and Soccer

The number of students in the US participating in high school basketball and soccer has steadily increased over the past few years.

basketball and soccer

The following table shows some information about this.

High School Sport Number of Students Participating in $\bm{2015}$
(Thousands) 		Average Rate of Increase
(Thousands per Year)
Basketball
Soccer

Consider the above table to answer the following questions.

a Write two equations to represent the situation.
b Graph the equations from Part A in the same coordinate plane. Use the graph to predict the approximate year when the number of students participating in these two sports will be the same.
Explore

Lines on the Same Plane

In the graph below, four lines and their corresponding linear equations can be seen on a coordinate plane. Investigate which lines intersect at one point, which lines intersect at infinitely many points, and which lines do not intersect at all.

lines
Discussion

System of Equations

Two important concepts will be discussed here.

Concept

Equation In Two Variables

An equation in two variables is a mathematical relation between two equal quantities that involves two variables.
Equations in two variables
Solving an equation in two variables results in an ordered pair that makes the equation true.
Concept

System of Equations

A system of equations is a set of two or more equations involving the same variables. The solutions to a system of equations are values for these variables that satisfy all the equations simultaneously. A system of equations is usually written as a vertical list with a curly bracket on the left-hand side.
Graphically, solutions to systems of equations are the points where the graphs of the equations intersect. For this reason, these solutions are usually expressed as coordinates.
The graph of a system of equations.

Systems of equations can be solved graphically or algebraically.

A system that contains only linear equations is called a linear system.
Discussion

Solving a System of Linear Equations Graphically

Solving a system of linear equations graphically means graphing the lines represented by the equations of the system and identifying the point of intersection. Consider an example system of equations.
To solve the system of equations, three steps must be followed.
1
Write the Equations in Slope-Intercept Form
expand_more
Start by writing the equations in slope-intercept form by isolating the variables. For the first linear equation, divide both sides by For the second equation, add to both sides.
▼
Solve for
DivEqn

WriteSumFrac

Write as a sum of fractions

MovePartNumRight

MoveNegNumToFrac

Put minus sign in front of fraction

CalcQuot

Calculate quotient

IdPropMult

Identity Property of Multiplication

▼
Solve for
AddEqn

RearrangeEqn

Rearrange equation

2
Graph the Lines
expand_more

Now that the equations are both written in slope-intercept form, they can be graphed on the same coordinate plane.

Graphs of two lines using the slopes and y-intercepts
3
Identify the Point of Intersection
expand_more

The point where the lines intersect is the solution to the system.

Point of intersection

The lines appear to intersect at Therefore, this is the solution to the system — the value of is and the value of is

Sometimes the point of intersection of the lines is not a lattice point. In this case, the solution found by solving the system of equations graphically is approximate.
Example

Modeling With a System of Linear Equations

Tearrik is throwing a party and bought some sodas and chips.

Sodachips.jpg

He bought items and spent The cost of a soda is and the cost of a bag of chips is This can be modeled by a system of equations.
Here, is the number of sodas and the number of bags of chips Tearrik bought. Solve the system by graphing and find how many sodas and bags of chips he bought.

Hint

Start by writing both linear equations in slope-intercept form.

Solution
To start, each equation in the system will be written in slope-intercept form. This means that the variable will be isolated in both equations.
▼
Write in slope-intercept form
SubEqn

SubEqn

DivEqn

WriteSumFrac

Write as a sum of fractions

MovePartNumRight

MoveNegNumToFrac

Put minus sign in front of fraction

CalcQuot

Calculate quotient

Now, the slope and the intercept of each line will be used to draw the graphs on the same coordinate plane. Since the number of items cannot be negative, only the first quadrant will be considered for the graph.
lines

Finally, the point of intersection can be identified.

point of intersection

The point of intersection of the lines is In the context of the situation, this means that Tearrik bought sodas and bags of chips.

Pop Quiz

Finding the Solution to a System of Equations by Graphing

The lines that correspond to a system's equations are graphed below. What is the solution to the system?

system with one solution
Discussion

Consistent and Independent Systems

The system of equations in the previous example had one solution. This leads to two important definitions.

Concept

Consistent System

A system of equations that has one or more solutions is called a consistent system. For example, consider the following linear systems.
To determine the number of solutions, each system can be graphed on a coordinate plane.
In the graph, it can be seen that the first system has exactly one solution. The second system has infinitely many solutions. Therefore, since both systems have one or more solutions, they are consistent systems.
Concept

Independent System

An independent system is a system of equations with exactly one solution. Consider the following linear system.
To state the number of solutions, both lines of the system can be graphed on the same coordinate plane.
Since the system has exactly one solution, it is an independent system.
Example

Finding the Solution and Classifying a System of Equations

At Tearrik's party, Ignacio found one of Tearrik's homework assignments. He does not want to do Tearrik's homework for him, but Ignacio decides to quiz himself using the assignment.

notebook and homework

Help Ignacio determine whether the given systems of equations are consistent systems and whether they are independent systems.

a
b

Hint
a Solve the system by graphing and determine the number of solutions.
b Solve the system by graphing and determine the number of solutions.

Solution
a The system will be solved graphically. To do so, both linear equations must to be expressed in slope-intercept form. In the first equation, the variable is already isolated. Therefore, only the second equation needs to be rewritten.
▼
Write in slope-intercept form
SubEqn

DivEqn

WriteDiffFrac

Write as a difference of fractions

DivNegNeg

MovePartNumRight

MoveNegDenomToFrac

Put minus sign in front of fraction

SubNeg

CalcQuot

Calculate quotient

Now that both linear equations are expressed in slope-intercept form, their slopes and intercepts can be used to draw the lines on the same coordinate plane.
lines

The lines intersect at exactly one point.

intersection

Since the system has one or more solutions, it is a consistent system. Furthermore, since it has exactly one solution, it is also an independent system.

b Again, the system of equations will be solved graphically. To do so, both equations will be rewritten in slope-intercept form.
▼
Write in slope-intercept form
SubEqn

SubEqn

DivEqn

WriteSumFrac

Write as a sum of fractions

MoveNegNumToFrac

Put minus sign in front of fraction

MovePartNumRight

CalcQuot

Calculate quotient

IdPropMult

Identity Property of Multiplication

Note that both equations are simplified to the same equation. It can be graphed by using its slope and its intercept.
line
The lines overlap each other. Therefore, they intersect at infinitely many points. Since the system has one or more solutions, it is a consistent system. However, since it does not have exactly one solution, it is not an independent system.
Discussion

Dependent and Inconsistent Systems

In the last example, one of the systems had infinitely many solutions. Systems of equations with infinitely many solutions have a special name.

Concept

Dependent System

A dependent system is a system of equations with infinitely many solutions. Consider the following linear system.
To find the number of solutions, both lines of the system can be graphed on the same coordinate plane.
The lines overlap each other. Therefore, the system has infinitely many solutions. This means that this is a dependent system.

Furthermore, systems with no solutions also have a special name.

Concept

Inconsistent System

A system of equations that has no solutions is called an inconsistent system. For example, consider the following linear system.
To determine the number of solutions, the system can be graphed on a coordinate plane.
The lines do not intersect. The system has no solution and is therefore an inconsistent system.
Example

Finding the Solution and Classifying a System of Equations

Mark cannot go to Tearrik's party because he has recently started working with his father at a car dealership. At the dealership, Mark's father sells sedans and trucks.

cars and trucks
Mark modeled the number of sedans and trucks sold with a system of equations.
Here, represents the number of days since Mark started working in the agency. In Equation (I) the variable represents the number of sedans sold. Similarly, in Equation (II) represents the number of trucks sold. Mark wants to interpret this system of equations in terms of consistency and independence. Help him do this!

Hint

Graph both equations on the same coordinate plane.

Solution
To interpret the system in terms of consistency and independence, both linear equations will be drawn on the same coordinate plane. To do so, the slope-intercept form will be used. Since the first equation is already written in this form, the variable will be isolated in the second equation.
▼
Write in slope-intercept form
AddEqn

DivEqn

WriteSumFrac

Write as a sum of fractions

MovePartNumRight

CalcQuot

Calculate quotient

Now that both equations are written in slope-intercept form, their slopes and intercepts can be used to draw the lines.
lines

Next, recall the definitions of consistent, inconsistent, dependent, and independent systems.

Consistent System A system of equations that has one or more solutions.
Independent System A system of equations with exactly one solution.
Dependent System A system of equations with infinitely many solutions.
Inconsistent System A system of equations that has no solution.

With this information in mind, the lines will be considered one more time.

lines

Since the lines do not intersect each other, the system has no solution. Therefore, it is an inconsistent system.

Pop Quiz

Classifying Systems

A system of equations can be consistent or inconsistent. In addition, a consistent system can be independent or dependent. Classify the following systems of equations in terms of consistency and independence.

system with one solution
Closure

Basketball and Soccer

With the content learned in this lesson, the challenge presented at the beginning can be solved. It has been said that the number of students in the US participating in high school basketball and soccer has steadily increased over the past few years.

basketball and soccer

The following table shows some information about this.

High School Sport Number of Students Participating in $\bm{2015}$
(Thousands) 		Average Rate of Increase
(Thousands per Year)
Basketball
Soccer

Consider the above table to answer the following questions.

a Write two equations in slope-intercept form to represent this situation.
b Graph the equations from Part A in the same coordinate plane. Use the graph to predict the approximate year when the number of students participating in these two sports will be the same.

Hint
a For both equations, let be the number of years since
b Pay close attention to the coordinate of the point of intersection of the lines.

Solution
a To start, the equation for the number of students participating in high school basketball will be written. The variables can be defined as follows.
Variable Meaning of the Variable
Number of years since
Number of students that participate in high school basketball (thousands)
From the given table, it is known that in the year thousands students participated in high school basketball. Also, the yearly average rate of increase is thousand students. With this information, an equation can be written.
Next, the equation for the number of students participating in high school soccer will be written. The variables can be defined in a similar way as before.
Variable Meaning of the Variable
Number of years since
Number of students that participate in high school soccer (thousands)
Again, it is seen in the table that in the year thousands students participated in high school soccer. Also, the yearly average rate of increase is thousand students. With this information, the second equation can be written.
b The equations found in Part A form a system of equations.
Since both lines are already written in slope-intercept form, they can be graphed on the same coordinate plane.
lines

The lines intersect at exactly one point. Therefore, this is a consistent and independent system. The solution is and This means that years after — the year — there will be one million high school students participating in both sports.


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