Topics
3. Systems of Linear Equations & Inequalities
topic badge
United States of AmericaTennessee
Integrated Math 1

3.01 Solving systems of equations by graphing

Lesson
Practice
Lesson

Concept summary

A system of equations is a set of equations which have the same variables.

A solution to a system of equations is any set of values of all variables in that system which is a solution to each equation in the system.

A solution can also be thought of graphically as the point(s) of intersection of the graphs of the equations (the points in common to all graphs):

A four quadrant coordinate plane is drawn with two parallel lines leaning to the left.

When two lines are parallel and distinct, they have no points of intersection. The corresponding system of equations has no solutions.

A four quadrant coordinate plane is drawn with a dashed line of positive slope passing through the origin. The line is composed of two alternating colors of dashes and dots representing two lines of the same slope, and solutions.

When two lines are identical, they intersect at every point. The corresponding system of equations has infinitely many solutions.

A four quadrant coordinate plane with two lines drawn intersecting each other at a common point. The lines intersect at a point located at (2, negative 1)

When two lines are not parallel, they have exactly one point of intersection. The corresponding system of equations has one solution.

A solution to a system of equations in a given context is said to be viable if the solution makes sense in the context, and non-viable if it does not make sense within the context, even if it would otherwise be algebraically valid.

Worked examples

Example 1

Consider the system of two equations shown in the graph:

-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
a

How many solutions does this system of equations have?

Approach

The solution(s) to a system of equations can be represented graphically as their point(s) of intersection.

Solution

This system has one point of intersection and therefore one solution.

Reflection

This system consists of two linear equations. Is it possible for a pair of linear equations to have more than one solution? Is it possible for a system of two linear equations to have no solutions?

b

Determine the solution to the system of equations as an ordered pair \left(x, y\right).

Solution

The point of intersection occurs at \left(-4, -2\right).

Example 2

Tyson is saving money in order to purchase a new smart-phone for \$800 when the latest model is released. He currently has \$350 saved up, and is able to put away \$100 each month.

a

Write a system of equations to represent the situation.

Approach

To write a system of equations, we will need to define some variables. Let's choose y to represent an amount of money (in dollars), and x to represent the number of months that have passed.

Solution

Using these variables, the amount of money Tyson has saved over time can be represented by y = 350 + 100x. The price of the smart-phone can be represented by y = 800.

b

Sketch the two lines representing these equations on the coordinate plane.

Approach

All of the values involved in the question are multiples of \$50, so we can use this for the scale of the y-axis. Also, both x and y only make sense for positive values in this context, so we only need to think about the first quadrant.

Solution

-1
1
2
3
4
5
6
7
8
x
100
200
300
400
500
600
700
800
900
y
c

If the new phone is to be released in 5 months time, determine if Tyson will be able to afford it on release.

Solution

The point of intersection on the graph occurs at (4.5, 800), meaning that in 4.5 months Tyson will have saved \$800. Therefore Tyson will have saved enough money before the phone is released.

Reflection

While the model equation of Tyson's savings is linear, in reality he probably puts money away once per month or once per week depending on how often he gets paid.

So although the point of intersection is at x = 4.5 months, Tyson might not actually reach \$800 in savings until the end of the 5th month.

Outcomes

M1.N.Q.A.1

Use units as a way to understand real-world problems.*

M1.N.Q.A.1.A

Choose and interpret the scale and the origin in graphs and data displays.*

M1.A.CED.A.3

Create individual and systems of equations and/or inequalities to represent constraints in a contextual situation, and interpret solutions as viable or non-viable.*

M1.A.REI.C.3

Write and solve a system of linear equations in real-world context.*

M1.A.REI.D.5

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x). Find approximate solutions by graphing the functions or making a table of values, using technology when appropriate.*

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP5

Use appropriate tools strategically.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

What is Mathspace

About Mathspace