T456
Mathematics Success – Grade 8
LESSON 18: Solving Systems Algebraically
[OBJECTIVE]
The student will solve systems of linear equations algebraically in mathematical and
real world situations.
[PREREQUISITE SKILLS]
solving equations
[MATERIALS]
Student pages S221 – S235
Two different colored pencils
[ESSENTIAL QUESTIONS]
1. Explain how to determine which variable to solve for when solving a system of
equations algebraically.
2. When solving a system of equations algebraically, what outcome indicates that
there is no solution? Justify your answer.
3. When solving a system of equations algebraically, what outcome indicates that
there are infinite solutions? Justify your answer.
[WORDS FOR WORD WALL]
system of linear equations, substitution
[GROUPING]
Cooperative Pairs (CP), Whole Group (WG), Individual (I)
*For Cooperative Pairs (CP) activities, assign the roles of Partner A or Partner B to
students. This allows each student to be responsible for designated tasks within the
lesson.
[LEVELS OF TEACHER SUPPORT]
Modeling (M), Guided Practice (GP), Independent Practice (IP)
[MULTIPLE REPRESENTATIONS]
SOLVE, Verbal Description, Pictorial Representation, Concrete Representation,
Graphic Organizer. Algebraic Formula
[WARM-UP] (IP, I, WG) S221 (Answers on T465.)
Have students turn to S221 in their books to begin the Warm-Up. Students will solve
equations. After students have completed the Warm-Up, review the solutions as a
group. {Graphic Organizer, Algebraic Formula}
[HOMEWORK]
Take time to go over the homework from the previous night.
[LESSON] [1 – 2 Days (1 day = 80 minutes) – M, GP, WG, CP, IP]
Mathematics Success – Grade 8
T457
LESSON 18: Solving Systems Algebraically
SOLVE Problem
(WG, GP) S222 (Answers on T466.)
Have students turn to S222 in their books. The first problem is a SOLVE problem.
You are only going to complete the S step with students at this point. Tell students
that during the lesson they will learn how to solve systems of equations algebraically.
They will use this knowledge to complete this SOLVE problem at the end of the
lesson. {SOLVE, Verbal Description, Graphic Organizer}
Solving Systems Algebraically – One Standard, One Slope-intercept Form
(M, GP, IP, CP, WG) S222, S223, S224, S225 (Answers on T466, T467, T468, T469.)
M, GP, CP, WG:
Have students turn to S222. Students will begin to analyze
equations and apply the knowledge they acquired from
solving systems by graphing. Students will transition into
solving systems of equations algebraically. Assign the roles
of Partner A and Partner B. {Verbal Description, Graphic
Organizer, Algebraic Formula}
MODELING
Solving Systems Algebraically – One Standard, One Slope-intercept Form
*Teacher Note: As you model solving equations algebraically, colors can be one
of the most helpful tools for students to see exactly what is being substituted into
an equation. Students will box different portions of the equations to also assist
with this, but having two colors to use when writing out the equations will be very
helpful for students to clearly see the substitution.
Step 1: Direct students’ attention to Question 1 on S222.
• Partner A, what do you notice about the equations above? (The first
one is in slope-intercept form and the second one is not.) Record.
• Partner B, what does it mean to solve a system of linear equations
like the two shown above? (It means that we are finding a solution
of an x-coordinate and y-coordinate that can be substituted into both
equations and make the statements true.) Record.
• Partner A, what method have we used to solve systems of equations in
the past? [We graphed both lines and found the solution(s).] Record.
• Partner B, what are the possible outcomes when solving a system of
equations? (We can have one solution, no solution or infinite solutions.)
Record.
• Partner A, why might the system of equations at the top of the page
be more challenging to graph? (Only the first equation is in slopeintercept form, so the second equation would require an adjustment
into slope-intercept form before graphing.)
• Can you think of any other reasons why it might be beneficial for us to
have an algebraic way of solving for solutions rather than graphing?
(Answers will vary. One point that should be explained is that when
T458
Mathematics Success – Grade 8
LESSON 18: Solving Systems Algebraically
graphing, the intersection may be at a point where the coordinates are
not as easily identified, as in a fraction of a unit, or a number that is so
great that plotting the point would take an excessive amount of time.)
Step 2: Direct students’ attention to the top of S223.
• Another option we have is to solve the system of equations algebraically
(by substitution) instead of graphically. Use the steps below to help
you solve the system.
• Partner B, read Step 1. Choose one equation to begin. (Suggestion:
Choose the equation that has a variable that is already isolated.)
• Partner A, do either of the equations have a variable that is isolated
or close to being isolated? (Yes, y = x – 50) Have students circle this
equation so they know the equation we are beginning with and pick a
color to use.
• Partner A, read Step 2. Is a variable isolated? If yes, continue to Step
3. If no, isolate one of the variables.
• Partner B, in the equation we chose, is a variable isolated? (Yes, y is
isolated in y = x – 50.) Record.
Step 3: Partner B, read Step 3. Box the expression that is equal to the isolated
variable.
• Partner A, what expression should we box? (Place a box around x –
50.) Have students draw the box.
• Partner A, read Step 4. Using the other equation, substitute the boxed
expression for the isolated variable and solve for the only variable
left.
• Partner B, what is the other equation? (x + 2y = 200)
• Partner A, for what variable are we substituting the boxed expression?
Explain your thinking. (The y is the isolated variable.)
• Begin by having the students write the second equation in a different
color. As they write the second line where they are substituting,
have them write everything in the second color except for the
boxed expression which will replace y. Use the original color for this
expression.
• Partner B, what will the new equation be when we substitute the
boxed expression? [x + 2(x – 50) = 200] Record.
• Be sure that students place the boxed expression inside of parentheses.
This is important to be sure that any coefficient is multiplied by the
whole expression that is substituted.
Step 4: Partner A, what do you notice about the equation now? (There is only
one variable left, which is x.)
• Partner B, how do we continue solving for x? (Distribute 2 to every
term inside of the box or parentheses.)
Mathematics Success – Grade 8
T459
LESSON 18: Solving Systems Algebraically
• Partner A, what is the simplified equation after you distribute?
(x + 2x – 100 = 200) Record.
• Partner A, what is the next step? (Combine like terms.)
• Partner B, what is the simplified equation after combining like terms?
(3x – 100 = 200) Record.
• Partner B what is the next step? (Add 100 to each side) Record.
• Partner A, what is the final step? (Divide each side by 3.) Record.
• Partner B, what is value of x? (x = 100) Record.
• Partner A, are we finished solving this system of equations? (No,
because we only found the value of x and our answer to a system
must be an ordered pair.)
• Partner A, read Step 5. Using either of the original equations, substitute
the known value and solve for the unknown value.
Step 5: Have student pairs discuss this question. Why do you think we can
substitute the x-value into either of the equations to solve for y? (The
point of solving a system of equations is to find the solution or where
the graphs intersect. Most likely, this solution will require the same
x-coordinate and the same y-coordinate. Substituting a known value,
which the two equations should share, can take place with either equation
because they should both yield the same y-value. If they do not, then we
need to find our mistake.)
• Partner B, which equation should we use? (y = x – 50)
• Partner A, what are we substituting? (We will replace x with the value
of 100.) Have students complete this step.
• Partner B, what is the value of y if we replace x with 100? (100 – 50
= 50, therefore y = 50.) Record.
• Partner B, what is Step 6? Check your solution and write your answer.
• Partner A, what is the solution? (100, 50) Record.
• Partner B, substitute (100, 50) into the equation y = x – 50, while
Partner A, substitutes the same coordinates into x + 2y = 200.
• Partner B, what did you find? (50 = 100 – 50, which means 50 = 50.)
Did it check? (Yes, the solution works for the first equation.)
• Partner A, what did you find? (100 + 2(50) = 200, so 100 + 100 =
200, or 200 = 200) Did it check? (Yes, the solution works for the
second equation.)
Step 6: Direct students’ attention to the top of S224.
• Guide students through the problem on S224 using the same
questioning as the first question from S223.
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Mathematics Success – Grade 8
LESSON 18: Solving Systems Algebraically
IP, CP, WG:
Have students complete Questions 1 and 2 on S225. Have
students refer to S223 or S224 for the steps if they get
stuck. They should follow the same process. Regroup after
students have had a chance to complete these problems
and review the solutions. {Verbal Description, Graphic
Organizer. Algebraic Formula}
Solving Systems Algebraically – Both Standard Form
(M, GP, IP, CP, WG) S226, S227, S228 (Answers on T470, T471, T472.)
M, GP, CP, WG:
Have students turn to S226 in their books. Students will
complete the same type of activity as on S223 and S224,
but they will be introduced to a new scenario where both
equations will be in standard form. This will require them to
manipulate one of the equations to isolate a variable. Be sure
that students know their designation as Partner A or Partner B.
{Algebraic Formula, Verbal Representation, Graphic Organizer}
MODELING
Solving Systems Algebraically – Both Standard Form
Step 1: Direct students’ attention to the top of S226.
• Have students begin by looking at the equations that are under Step 1.
What do you notice about these two equations? (Both of the equations
are in standard form.)
• Partner A, what is Step 1? Choose one equation to begin. (Suggestion:
Choose the equation that has a variable that is already isolated or
close to being isolated.)
• Partner B, do either of the equations have a variable that is isolated
or close to being isolated? (Yes, x + 2y = 3) Have students circle this
equation so they know the equation with which we are beginning and
choose a color to use.
Step 2: Partner B, read Step 2. Is a variable isolated? If yes, continue to Step 3.
If no, isolate one of the variables.
• Partner A, in the equation we chose, is a variable isolated? (No.)
• Partner B, identify the easiest variable to isolate in the equation we
chose. (x is the easiest because we would simply subtract 2y from
each side of the equation and we could then move forward.)
• Partner A, what is the new equation after we isolate x? (x = 3 – 2y)
Record the isolation of x.
• Partner B, read Step 3. Box the expression that is equal to the isolated
variable.
• Partner A, what expression should we box? (Place a box around
3 – 2y.) Have students draw the box.
Mathematics Success – Grade 8
T461
LESSON 18: Solving Systems Algebraically
• Partner B, read Step 4. Using the other equation, substitute the boxed
expression for the isolated variable and solve for the only variable left.
• Partner A, what is the other equation? (2x + 3y = 1)
• Partner B, we are substituting the boxed expression for what variable?
(x)
Step 3: Begin by having the students write the second equation in a different
color. As they write the second line where they are substituting, have
them write everything in the second color except for the boxed expression
which will replace x. Use the original color for this expression.
• Partner A, what will the new equation be when we substitute the
boxed expression? [2(3 – 2y) + 3y = 1] Record.
• Be sure that students place the boxed expression inside of parentheses.
This is important to be sure that any coefficient is multiplied by the
whole expression that is substituted.
• Partner B, what do you notice about the equation now? (There is only
one variable left, which is y.)
• Partner A, how do we continue solving for y? (Distribute 2 to every
term inside of the box or parentheses.)
• Partner B, what is the simplified equation after you distribute?
(6 – 4y + 3y = 1) Record.
Step 4: Have students discuss the next step. (Combine like terms.)
• Partner B, what is the simplified equation after combining like terms?
(6 – y = 1) Record.
• Partner A, what is the next step? (Subtract 6 from each side)
Record.
• Partner B, what is the final step? (Divide each side by -1) Record.
• Partner A, what is value of y? (y = 5) Record.
• Partner B, are we finished solving this system of equations? (No,
because we only found the value of y and our answer to a system
must be an ordered pair.)
• Partner B, read Step 5. Using either of the original equations, substitute
the known value and solve for the unknown value.
• Partner A, which equation should we use? (x + 2y = 3)
• Partner B, what are we substituting? (We will replace y with the value
of 5.) Have students complete this step.
• Partner A, what is the value of x if we replace y with 5? (x + 2(5) =
3, simplifies to x + 10 = 3. We can subtract 10 from each side and,
x = -7.) Record.
• Partner A, what is Step 6? Check your solution and write your answer.
• Partner B, what is the solution? (-7, 5) Record.
• Partner A, substitute (-7, 5) into the equation x + 2y = 3, while
Partner B, substitutes the same coordinates into 2x + 3y = 1.
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Mathematics Success – Grade 8
LESSON 18: Solving Systems Algebraically
• Partner A, what did you find? [-7 + 2(5) = 3 simplifies to
-7 + 10 = 3, or 3 = 3.] Did it check? (Yes, the solution works for the
first equation.)
• Partner B, what did you find? [2(-7) + 3(5) = 1, so -14 + 15 = 1, or
1 = 1] Did it check? (Yes, the solution works for the second equation.)
Step 5: Direct students’ attention to the top of S227.
• Guide students through the problem on S227 using the same
questioning as the first question from S226.
IP, CP, WG:
Have students complete Questions 1 – 2 on S228. Have
them refer to S226 or S227 for the steps if they need
support. They should follow the same process but without
using the organizer for steps. Regroup after students have
had a chance to complete these problems and review the
solutions. {Verbal Description, Graphic Organizer, Algebraic
Formula}
Solving Systems Algebraically – No Solution and Infinite Solutions
(M, GP, IP, CP, WG) S229, S230 (Answers on T473, T474.)
M, GP, CP, WG:
Have students turn to S229 in their books. Students will
explore two new situations that have scenarios where the
systems have no solution or infinite solutions. Be sure
students know their designation as Partner A or Partner B.
{Verbal Description, Graphic Organizer, Algebraic Formula}
MODELING
Solving Systems Algebraically – No Solution and Infinite Solutions
Step 1: Direct students’ attention to Questions 1 and 2.
• Assign Partner A to complete Question 1 while Partner B completes
Question 2. Give students a moment to solve the systems and discuss
with partners.
• Partner A, what did you notice about Question 1? (In solving, students
should arrive at a roadblock where their equation simplifies to a false
statement of 3 ≠ 5.) Let’s go back and review the work.
• What equation did you choose to solve and box? (2x + y = 3 is easily
isolated so that y = 3 – 2x.)
• Where are we substituting 3 – 2x? (We substitute it in for y in 2x + y
= 5.)
• What happens when we simplify after substituting? (The variable
terms of 2x cancel and we are left with 3 ≠ 5.)
• Let’s take a look at Question 2.
Mathematics Success – Grade 8
T463
LESSON 18: Solving Systems Algebraically
Step 2: Direct students’ attention to Question 2.
• Partner B, what did you notice about Question 2? (In solving, students
should arrive at a roadblock where their equation simplifies to a true
statement of 3 = 3.) Let’s go back and review the work.
• What equation did you choose to solve and box? (x + y = 1 is easily
isolated so that y = 1 – x.)
• Where are we substituting 1 – x? (We substitute it in for y in 3x + 3y
= 3.)
• What happens when we simplify after substituting? (The variable
terms of 3x cancel and we are left with 3 = 3.)
Step 3: Direct students’ attention to the statement below Questions 1 and 2.
• Have students pairs change all of the equations from Question 1 and
2 into slope-intercept form.
• Partner A, what are the equations for Question 1 in slope-intercept
form? (y = -2x + 3 and y = -2x + 5) Record.
• Partner B, what are the equations for Question 2 in slope-intercept
form? (y = -x + 1 and y = -x + 1) Record.
• Partner A, what do you notice about the slope for the equations in
Question 1? (They are both -2.) Record.
• What do you notice about the y-intercepts for the equations in Question
1? (They are different.) Record.
• What can you conclude about these lines? (They are parallel lines and
have no solution.) Record.
• Partner B, what do you notice about the slope for the equations in
Question 2? (They are both -1.) Record.
• What do you notice about the y-intercepts for the equations in Question
2? (They are both 1.) Record.
• What can you conclude about these lines? (They are identical lines
with infinite solutions.) Record.
IP, CP, WG:
Have students solve Questions 9 and 10 on S230. Students
will practice solving systems where the outcome will be no
solution or infinite solutions. Be sure students understand
the form that they will see when the result is either of
these situations and how to prove that they are indeed
parallel or identical lines if they must. {Verbal Description,
Graphic Organizer, Algebraic Formula}