John Mowbray
Algebra 1
Algebra 1 – Solving Systems of Linear Equations
STANDARDS:
ESSENTIAL QUESTION:
The content focus is Systems of Linear Equations.
 HSA-CED.A.3: Represent constraints by equations or
inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or
nonviable options in a modeling context.
 HSA-REI.C.5: Prove that, given a system of two
equations in two variables, replacing one equation by
the sum of that equation and a multiple of the other
produces a system with the same solutions.
 HSA-REI.C.6: Solve systems of linear equations
exactly and approximately (e.g., with graphs),
focusing on pairs of linear equations in two variables.
 Can you solve a system of linear equations?
What are the MOST
important concepts or skills?

Students needs to be able to solve a system of linear
equations from a naked math perspective and from word
problems.
ACTIVATING STRATEGY:


Students will take the pre-assessment.
Once completed we will discuss the topic and what
systems of equations are.
How will you activate your
lesson or link to prior
knowledge?
TEACHING STRATEGIES:
What instructional strategies
will you use in your lesson?
-Complete student notes packet with class showing multiple
examples with explanations.
-Complete Independent Practice Packet that allows them to
work through various examples. Students will use
technology such as Desmos, WolframAlpha, TI-84 graphing
calculators, and a learning game called Kahoot! to complete
and review the material.
-Desmos is an online graphing program that will allow the
students to solve a system of equations using graphing and
be able to identify the solution. This will help students with
the first part of the activity packet. Students can also use
this on the assessment. www.desmos.com
-WolframAlpha is an online math application that gives help
to students solving equations or any other math questions
John Mowbray
Algebra 1
that they may have. Student can use this at any point while
they are working. www.wolframalpha.com
-Students will also look at videos using Khan Academy to
further assist them with explanations while they are working
instead of just waiting for teacher explanations.
https://www.khanacademy.org/math/algebra/systems-ofeq-and-ineq
-Teacher and assistants will work with small groups and they
are working on independent practice to make sure everyone
ends up with an equal understanding of the topic.
Math Game (To be used towards end of class after
independent practice. If not used then, try the beginning of
the following class depending on length of classes).
-Teacher sets up account at getkahoot.com and locates or
creates review game for class.
-Have computer lab access, class set of laptops in classroom
or allow students to use smartphones or tablets.
-Students login and create a username for Kahoot.it
-Teacher puts the review game on main computer and
project the questions on whiteboard.
-Students have around two minutes to answer each
question.
-Students answer via laptops, smartphones, or tablets. Only
username are shown to assure privacy, and students are
ranked based off their answers.
-When students are finished pull up response sheet to see
how students did and to check for a need for re-teaching.
-There is also a smartphone app that allows students to
answer quicker than using the website on their phones.
Website:
https://play.kahoot.it/#/?quizId=efe2d804-0931-45e0-b53acbb13f541b68
SUMMARIZING STRATEGIES:
How will students summarize
what they are learning during
the lesson and at the end?

Students will complete the post assessment to see if
there was a growth in student understanding.
John Mowbray
Algebra 1
Student Activity Packet
Definition(s):
 A system of equations is a set of equations dealt with simultaneously for which a common
solution, if possible, is sought.
 A solution to system of equations is a point that lies on the graph of each equation in the system.
Part I: Identifying the solution(s) to a system of equations
Use Desmos to complete the following part of the assessment.
1. The equations y = x + 1 and y = ½x + 2 are graphed below to the right. Do the graphs intersect? If
so, name the point of intersection.
2. Verify your intersection point is correct by substituting the coordinates of the intersection point in
for x and y into the original equations. Show your work and explain how your work verifies that
the intersection point is correct
3. The intersection point (2, 3) is the solution to the system of equations in question 1. To solve a
system of equations graphically, graph both equations and see where they intersect. The
intersection point is the solution.
4. Solve Graphically. Graph the following system of equations on your graphing calculator. Paste a
screen shot of your graphs in the space below. (Hint: You will need to solve both equations in
terms of y to graph the equations on your graphing calculator. Show your work for completing
this.)
4x-6y=12
2x+2y=6
y1 = _______________________
y2= _______________________
5. What does the solution to the system of equations from question 4 appear to be? Verify your
intersection point is correct by substituting the coordinates of the intersection point in for x and y
John Mowbray
Algebra 1
into the original equations. Show your work and explain how your work verifies that the
intersection point is correct.
6. On your graphing calculator, graph the following system of equations and paste a screen shot of
your graphs in the space below.
f1(x) = 2x - 5
f2(x) = -¼x - 1
7. Is the exact solution to the system obvious? Why or why not?
8. When the solution contains non-integer values, the solution can be calculated graphically using
the intersection point(s) tool. Select 6: Points and Lines from the graphing menu and arrow
down to select 3: Intersection Point(s). Select the graph of f1(x) = 2x - 5 by hitting “enter” when
the line is flashing. Next, select the graph of f2(x) = -¼x – 1 by hitting “enter” when the line is
flashing. The solution will appear. Record the solution and paste a screen shot of graphs with the
solution shown below.
9. On a new screen, graph the following system of equations and paste of screen shot of the
graphs below. (Hint: you will need to solve the second equation for y to enter the equation on
your graphing calculator.)
y = 2x + 4
-4x + 2y = 6
10. What relationship exists between the lines? Does the system appear to have a solution? Why or
why not?
John Mowbray
Algebra 1
The system of equations from question 9 represents one of the three possible cases of systems of
equations, called an inconsistent system.
11. Graph the following systems of equations on your graphing calculator and complete the first two
columns of the chart below. If the system has a solution, give the coordinates of the solution.
(Caution: Lines that appear to be parallel are not always parallel. You may need to change your
viewing window to see a point of intersection.)
System of
Equations
1.
g(x) = 2x – 0.6
h(x) = -¼x
2.
i(x) = 0.5x + 9
j(x) = ½ x + (18/2)
3.
y=x–1
-x + y = -1
4.
k(x) = 1.5x + 6
l(x) = 3/2x - 4
Number of
Solutions
Coordinates of
solution, if one
exists
Case
Name of System
John Mowbray
Algebra 1
5.
-5x – y = 4
10x + 2y = 7
6.
m(x) = 4x + 9
n(x) = 3.8x - 8
Three Possible Cases: When we graph a system of two linear equations, one of three things may
happen.
Categorizing Systems by Names – Consistent, Inconsistent, Dependent, and Independent
1. The lines have one point of intersection. The point of intersection is the only solution of
the system. The system is consistent and independent.
2. The lines are parallel. If this is the case, there is no point that satisfies both
equations. The system has no solution. The system is inconsistent.
3. The lines coincide. Therefore, the equations have the same graph and every solution of
one equation is a solution of the other. There is an infinite number of solutions. The
system is consistent and dependent.
This information is summarized in the chart below.
Case
1. lines intersect
2. parallel lines
3. lines coincide
Number of Solutions
One
Zero
Infinitely many
Name of System
Consistent, independent
Inconsistent
Consistent, dependent
Information in question 11 is taken from http://www.jcoffman.com/Algebra2/ch3_1.htm
12. Using the information from above, complete the last two columns of the chart in question 12.
13. Summarize all definitions and concepts concerning systems of linear equations you have learned
thus far.
Part II: Solving Systems of Equations by the Substitution Method
John Mowbray
Algebra 1
You can solve a system of equations by solving one equation for one of the variables; then substitute this
result into the second equation.
Here is an example of solving a system of equations by substitution.
Find the solution to the following system of equations.
x=y+3
y + 3x = 1
Step 1: The first equation is solved for x. Substitute (y + 3) in for x in the second equation.
y + 3x = 1 becomes y + 3(y + 3) = 1
Step 2: Solve for y.
y + 3y + 9 = 1
4y = -8
y = -2
Step 3: Substitute -2 in for y in the first equation to solve for x.
x = y + 3 becomes
x = -2 + 3
x=1
Step 4: Write the solution as an ordered pair.
The solution to the system of equations is (1, -2).
15. Use substitution to solve the following system of equations:
3x + 4y = -1
6x – 2y = 3
Solve for x in the first equation. Show your work.
Substitute the value you got for x into the second equation, rewrite the equation, and solve for y.
(Hint: Remember to use parentheses.) Show your work.
Now, substitute the value you got for y into either equation and solve for x.
The solution to this system of equations is (x, y) = __________
Check your work by graphing the system of equations on your graphing calculator. Find the
intersection of the two lines. Does it agree with your answer above?
John Mowbray
Algebra 1
16. Solve each of the following systems.
a. Graph each system of equations.
b. Classify each system as consistent independent, inconsistent, or consistent dependent.
(State the solution if one exists.)
c. Solve the systems by substitution. Be sure to show your work.
d. Notice when solving by substitution in (b) and (c), either an untrue statement results or an
identity results. Compare these answers with the type of system you named in bullet 2.
a. 2x + 3y = 3
12x – 15y = -4
b. x + 3y = 0
2x + 6y = 7
c. 1.4x - .3y = 20
2.8x –. 6y = 40
You can classify a system of linear equations by the number of solutions. Systems that have a unique
solution are independent systems. However, not every system has a unique solution. Systems with an
untrue statement are inconsistent. Systems with an identity are consistent and dependent. Summarize
what types of solution you will get when solving a system by substitution by providing examples of each
type.
17. What will happen if you have a system of 3 equations with 3 variables?
3x – 5y + z = 9
x – 3y – 2z = -8
5x – 6y + 3z = 15
What variable would you solve for in the first equation? _____ What is your
result?_____________________
Now, use this result and substitute it into the second and the third equations. Do you notice that you now
have two equations with two variables? Solve this new system. Substitute the results into an equation to
get the value of the variable that you chose first. The solution to this system of equations is (x, y, z) =
_______________
John Mowbray
Algebra 1
Verify by substituting the solution into each of the equations.
Part III: Solving Systems of Equations by the Elimination Method
18. Let’s go back to the same system of equations that we used in #15 and find another method to solve
them.
3x + 4y = -1
6x – 2y = 3
Look at the second equation. What value could you multiply the second equation by so that when you add
the first equation to the new second equation, the sum of the y variables would be 0? ___
2( 6x – 2y = 3)______________________________________
Now add the two equations: 3x + 4y = -1
12x – 4y = 6
Solve for x. _____ Substitute this value into either equation and solve for y. Does your answer
correspond to the answer you got by the substitution method, and by graphing?
Could you have first found a value in the first equation to eliminate x? Look again at the first equation.
What value could it be multiplied by in order to eliminate the x variables when the two equations would
be added together?
Here is the work that another student has done to solve a system of equations. Explain what the student
did in each step.
5x + 4 y = 24
3x = 2 + 2y
Step 1. 5x + 4y = 24
3x – 2y = 2
John Mowbray
Algebra 1
Step 2. 15x + 12y = 72
-15x + 10y = -10
Step 3. 22 y = 62
Step 4. y = 31/11
Step 5. 3x = 2 + 2(31/11)
Step 6. x = 84/33
19. Solve each of the following systems by elimination. Check your answers by graphing, and paste a
screen shot of each graph with the solution shown.
a. 3x – 12y = 25
2x – 4y = 7
b. .5x = .25y
x + 2y = 30
John Mowbray
Algebra 1
Systems of Equations Pre and Post-Test
Write the system of equation and then solve the following:
1) Brad bought 2 trees and 5 bushes for $39. Steve bought 3 trees and 8 bushes for $61. What is the
cost of each tree?
2) Club A charges $50 plus a monthly fee of $75.
Club B charges $125 plus a monthly fee of $60.
After how many months would the cost for each club be the same?
3) If the price of 3 hats and 7 shirts is $189 and 5 hats and 11 shirts is $301, what is the price of
each shirt?
4) Paul bought 8 eating utensils for a total of $26. Spoons cost $4 and forks cost $3. How many did
he buy?
5) Steve has $80 and Rob has $145. Steve decides to save $6 of your allowance each week, while
Rob decides to spend $7 each week. After how many weeks will Steve and Rob have the same
amount of money?
6) The membership fee for joining a gardening association is $24 per year. A local botanical garden
charges members of the gardening association $3 for admission to the garden. Nonmembers of
the association are charged $6. After how many visits to the garden is the total cost for
members the same as the total cost for nonmembers?
John Mowbray
Algebra 1
Solve the following systems of equations.
7) Y=2x + 15
Y=4x – 7
8) 2x + 5y = 16
4x – 3y = 6
9) Y = 4x + 7
2x + 3y = 7
10) The graph below compares the income and expenses involved in the production and sales of tennis
shoes at a shoe factory.
How many pairs of tennis shoes must be sold for
income and expenses to be equal?
11) The graph below shows the amount of money Diana and Melinda each saved during their summer
vacation.
John Mowbray
Algebra 1
Which of these statements can be concluded from the graph?
A. Diana will always have more money than Melinda.
B.
Diana will never have more money than Melinda.
C.
The amount of Melinda’s savings becomes greater than
Diana’s after 12 weeks.
D. The amount of Diana’s savings becomes greater than
Melinda’s after 12 weeks.
12) Look at the system of equations below.
2
𝑦 = 𝑥 + 10
3
2
𝑦 = 𝑥 − 10
3
Which of these statements is correct?
A. The system has no solution.
B. The solution of the system is (-3, 8).
C. The solution of the system is (6, -6).
D. The system has an infinite number of solutions.
13) Look at the system of equations below.
12x−4y = 8
3x−y = 2
Which of these statements describes the graph of this system of equations?
A. the same line
B. two parallel lines
C. two lines that intersect only at (1,1)
D. two lines that intersect only at (0, −2)
14) Look at the system of equations below.
y = 3x - 2
2y = 6x – 4
Which of these describes this system?
A. two parallel lines
B. two equations of the same line
C. two lines that intersect only at (1, 1)
D. two lines that intersect only at (-1, -1)
John Mowbray
Algebra 1