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Some real-life situations can be modeled with more than one equation that can be joined into a system of equations. There are different methods of solving systems of equations. This lesson will focus on one of them, which involves graphing the equations.

Catch-Up and Review

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

Explore

Lines on the Same Plane

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

Four lines
Discussion

Introducing Systems of Equations

Consider the definition of equations in two variables.

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.
Two or more simultaneous equations in multiple variables form a system of equations.
Discussion

Systems 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.
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
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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
Solve for
2
Graph the Lines
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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
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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

Mark is throwing a party, so he bought some donuts and lollipops for his friends.

Donuts and lollipops
The number of donuts and lollipops that Mark has bought can be modeled by a system of equations.
Solve the system by graphing and find how many donuts and lollipops he bought.

Hint

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

Solution

To start, each equation in the system will be rewritten in slope-intercept form. This means that the variable will be isolated in both equations.
Write in slope-intercept form
Now, the slope and the intercept of each line will be used to draw the graphs on the same coordinate plane. The intercept of the first equation is so the first point is The slope is so the second point can be plotted by step to the right and step down on the graph. Since this graph uses a different scale, try moving steps right and steps down instead.
Line corresponding to the first equation

Since the number of items cannot be negative, only the first quadrant will be considered for the graph. The intercept of the second equation is so the first point on that line is The slope is To better match the scale of this graph, the second point can be plotted by going steps to the right and steps down.

Line corresponding to the second line

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 Mark bought donuts and lollipops.

Pop Quiz

Finding the Solution to a System of Equations

Consider the graph of a system of equations consisting of two lines. What is the solution to the system?

System with one solution
Discussion

Number of Solutions to a System of Linear Equations

When a system of linear equations has two equations and two variables, the system can have zero, one, or infinitely many solutions.

No Solution

If a system has no solution, its graph might look similar to this one.

Two parallel lines
Recall that the solution to a system is the point where the lines intersect. If a system has no solution, then the lines never intersect. In fact, the lines must be parallel, meaning that they have the same slope and different -intercepts. Here is an example of one such system.

One Solution

If a system of equations has one solution, its graph consists of two lines that intersect exactly once. The point of intersection is the solution to the system.

Two intersecting lines
In contrast to parallel lines, lines that intersect once must have different slopes. For example, the following system must have exactly one solution because the two lines have different slopes.

Infinite Number of Solutions

If a system of equations has infinitely many solutions, the lines intersect at infinitely many points. This means the lines lie on top of each other or coincide with each other.

Two coinciding lines

These lines are said to be coincidental, and since they have the same slope and intercept, they are different versions of the same line. Here is one example of a system that has an infinite number of solutions.

Example

Determining the Numbers of Tasks and Questions in Balloons

To start the party, Mark suggested they play a fun game. He invited everyone to a room full of balloons. Each balloon had a note with a task to complete or a question to answer.
Three balloons with notes inside of them
The number of tasks and questions can be determined by the following system of equations.
Solve the system of equations by graphing.

Hint

First, rewrite the equations into slope-intercept form. Then, graph the equations and find the point of intersection of the lines.

Solution

To solve the system of equations graphically, both equations should be rewritten in slope-intercept form first. Let be the independent variable and be the dependent variable.
Write in slope-intercept form

Put minus sign in front of fraction

Calculate quotient

Notice that both equations simplify to the same equation. This equation can be graphed by using its slope and its intercept.
Line

The lines overlap each other. They intersect at infinitely many points, which means that the system of equations has infinitely many solutions. This indicates that the system of equations was not set up properly. Maybe the same information was written in two different ways, which resulted in two equations that represent the same line.

Example

Scavenger Hunt

Later in the evening, Mark suggested doing a scavenger hunt. All the hunters were split into two teams. The team that found the most items would win.
The list of scavenger hunt items
External credits: @upklyak
Team the Adventure Seekers, found items. Team the Mystery Solvers, found items. Mark’s parents tallied up the scores and, to make it more of a challenge, decided to give the scores as a system of equations. Whoever solves the system will get an extra gift!
Solve the system using the graphing method to find how many items each team found.

Hint

Begin by rewriting the equations into slope-intercept form. Then, graph both equations on the same coordinate plane using their slopes and intercepts.

Solution

The given system of equations needs to be solved graphically.
To do so, both equations should be rewritten in slope-intercept form first. Let be the independent variable and be the dependent variable.
Write in slope-intercept form

Write as a sum

Put minus sign in front of fraction

Calculate quotient

Notice that the equations have the same slope of but different intercepts. This means that the lines are parallel. Graph them using the slope and the intercepts.
The graph of the first and second equations
Recall the the solution to the system of equations is the point of intersection of the lines. However, since the lines are parallel, they never intersect. Therefore, there is no solution to the system of equations. Since Mark's parents are not mathematicians, they probably made a mistake, so their system did not work.
Pop Quiz

Determining the Number of Solutions to a System of Equations

Consider the given system of equations. Does it have zero, one, or infinitely many solutions?

System with different number of solutions
Closure

Identifying the Number of Solutions From Slope-Intercept Form

This lesson focused on solving systems of linear equations by graphing. When equations in the system are written in standard or point-slope form, it is unclear how many solutions the system has. However, the situation is different when it comes to slope-intercept form.
If both equations of the system of equations are in slope-intercept form, there is no need to graph them in order to find the number of solutions. It can be found by analyzing the slopes and intercepts of the equations.
Characteristics Number of Solutions
Same slope and same intercept Infinitely many solutions
Same slope and different intercepts No solution
Different slopes One solution
These facts can be useful when solving systems of equations.
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