Unit 2 – Section 3
Solving Two-Step
Equations
Notes & Practice
Aligned to the Common
Core State Standards
7.EE.B.4
8th Grade Math Curriculum
Grades
6-9
Teacher’sGuide
CCSS7.EE.B.4
Usevariablestorepresentquantitiesinareal-worldormathematicalproblem,
andconstructsimpleequationsandinequalitiestosolveproblemsby
reasoningaboutthequantities.
a.Solvewordproblemsleadingtoequationsoftheformpx+q=randp(x÷q)
=r,wherep,q,andrarespeciKicrationalnumbers.Solveequationsofthese
formsKluently.Compareanalgebraicsolutiontoanarithmeticsolution,
identifyingthesequenceoftheoperationsusedineachapproach.Forexample,
theperimeterofarectangleis54cm.Itslengthis6cm.Whatisitswidth?
Objective
•  Studentswillbeabletosolvetwo-stepequations.
•  Studentswillbeabletosolveworldproblemsinvolvingtwo-step
equations.
DifferentiatedNotes
v
TraditionalNotes–Thesearebasicskeletalnotes.StudentsKillinthe
exampleswithyouanddothe“try”problemsindependently.
r
Modi/iedNotes–Thesecutdownontheamountofwritingthat
studentsarerequiredtodo.AllexamplesareKilledin,butpracticeproblems
areleftblankforstudentcompletion.(ExcellentforIEPand504
accommodationsinvolvingdifKicultieswithhearing,handwriting,etc.)
f
TeacherNotes–ThesearecompletelyKilledinforteacherreference.
Thesenotesarealsogreatforstudentswhohavebeenabsent.
DifferentiatedPractice(Homework)
v
AdvancedPractice–PerfectchallengeforyourhighKlyers!Many
problemsoverlapwithBasicPractice,butthisworksheetskipssomeofthe
basicproblemsandreplacesthemwithmoreadvancedpractice.
r
BasicPractice–Themajorityofyourstudentswillcompletethisonlevelassignment.
f
Modi/iedPractice–GreatoptionforstudentsrequiringamodiKied
amountofhomeworkperanIEPor504.Thisisacutdownversionofthe
BasicPracticeassignment(onepageonly).
Name: _______________________________________ Class: _______ Date: ____________
Solving Two-Step Equations
Steps for Solving a Two-Step Equation:
1)
2)
Example 1: Solve each equation. Show your work.
a) 2 b + 7 = −1
3
c)
Try:
m
− 9 = −5
4
5
a) −4 + g =1
4
c) 1 −
v
d
= −7
4
b) −6 + 5v = −21
d)
f −3
= 2.5
6
b) 7w + 2 = −19
d) p + 2.4 = 3.5
2
© Free to Discover (Amanda Nix) – 2016
Example 2: Use the Five-Step Method to solve each
word problem.
Five-Step Method
D
a) Jamie picks out a blouse to purchase. It is on
sale for 25% off plus she has a $10 off
coupon. She pays $23 for the blouse.
Determine the original price.
E
S
A
C
b) Michael is collecting sea glass from the beach.
He begins with 14 pieces in his collection and adds 3 new pieces per day. Determine how
many days it will take Michael to have 80 pieces of sea glass in his collection.
Try:
a) Kayla is given a monthly allowance. She puts 40% of the money directly into her savings
account. She spends $15 at the movies with her friends. At the end of the month, she has
$3 left. How much is her monthly allowance?
b) Juan has already run 6 miles this month. He plans to run an additional 2.5 miles per day.
At this rate, how long will take Juan to run 41 miles?
v
© Free to Discover (Amanda Nix) – 2016
Name: _______________________________________ Class: _______ Date: ____________
Solving Two-Step Equations
Steps for Solving a Two-Step Equation:
1)  Use reverse order of operations to undo one operation.
(This is usually addition or subtraction.)
2) Use the inverse operation to undo the second operation.
Example 1: Solve each equation. Show your work.
a)
2
b + 7 = −1
3
−7 − 7
3⎛2 ⎞
3
⎜ b ⎟ = (−8)
2⎝3 ⎠
2
b = −12
c) m − 9 = −5
4
+9 + 9
⎛m ⎞
4 ⎜ ⎟ = (4) 4
⎝4⎠
m =16
Try:
5
a) −4 + g =1
4
c) 1 −
r
d
= −7
4
b) −6 + 5v = −21
+6
+6
5v = −15
5
5
v = −3
d)
f −3
= 2.5
6
⎛f − 3⎞
6⎜
⎟ = (2.5) 6
⎝ 6 ⎠
f − 3 =15
+3 +3
f =18
b) 7w + 2 = −19
d) p + 2.4 = 3.5
2
© Free to Discover (Amanda Nix) – 2016
Example 2: Use the Five-Step Method to solve each
word problem.
a) Jamie picks out a blouse to purchase. It is on
sale for 25% off plus she has a $10 off
coupon. She pays $23 for the blouse.
Determine the original price.
p=original price
0.75p-10=23
p=44
The original price of the blouse is $44.
?
0.75(44)-10=23
23=23
Five-Step Method
D
Define the variable.
E
Write the equation.
S
Solve the equation.
A
Answer the question.
C
Check your solution.
b) Michael is collecting sea glass from the beach.
He begins with 14 pieces in his collection and adds 3 new pieces per day. Determine how
many days it will take Michael to have 80 pieces of sea glass in his collection.
?
d=days
14+3(22)=80
14+3d=80
80=80
d=22
It will take 22 days to have 80 pieces of sea glass.
Try:
a) Kayla is given a monthly allowance. She puts 40% of the money directly into her savings
account. She spends $15 at the movies with her friends. At the end of the month, she has
$3 left. How much is her monthly allowance?
b) Juan has already run 6 miles this month. He plans to run an additional 2.5 miles per day.
At this rate, how long will take Juan to run 41 miles?
r
© Free to Discover (Amanda Nix) – 2016
Name: _______________________________________ Class: _______ Date: ____________
2-3
Solving Two-Step Equations
Steps for Solving a Two-Step Equation:
1)  Use reverse order of operations to undo one operation.
(This is usually addition or subtraction.)
2) Use the inverse operation to undo the second operation.
Example 1: Solve each equation. Show your work.
a)
2
b + 7 = −1
3
−7 − 7
3⎛2 ⎞
3
⎜ b ⎟ = (−8)
2⎝3 ⎠
2
b = −12
c) m − 9 = −5
4
+9 + 9
⎛m ⎞
4 ⎜ ⎟ = (4) 4
⎝4⎠
m =16
Try:
5
a) −4 + g =1
4
+4
+4
4⎛5 ⎞
4
⎜ g ⎟ = ( 5)
5⎝4 ⎠
5
g =4
c)
f
d
= −7
4
−1
−1
⎛ d⎞
-4 ⎜ − ⎟ = (−8) (−4)
⎝ 4⎠
d = 32
1−
b) −6 + 5v = −21
+6
+6
5v = −15
5
5
v = −3
d)
b)
d)
f −3
= 2.5
6
⎛f − 3⎞
6⎜
⎟ = (2.5) 6
⎝ 6 ⎠
f − 3 =15
+3 +3
f =18
7w + 2 = −19
−2 −2
7w = −21
7
7
w = −3
p + 2.4
= 3.5
2
⎛ p + 2.4 ⎞
2⎜
⎟ = (3.5) 2
⎝ 2 ⎠
p + 2.4 = 7
−2.4 − 2.4
p = 4.6
© Free to Discover (Amanda Nix) – 2016
Example 2: Use the Five-Step Method to solve each
word problem.
a) Jamie picks out a blouse to purchase. It is on
sale for 25% off plus she has a $10 off
coupon. She pays $23 for the blouse.
Determine the original price.
p=original price
0.75p-10=23
p=44
The original price of the blouse is $44.
?
0.75(44)-10=23
23=23
Five-Step Method
D
Define the variable.
E
Write the equation.
S
Solve the equation.
A
Answer the question.
C
Check your solution.
b) Michael is collecting sea glass from the beach.
He begins with 14 pieces in his collection and adds 3 new pieces per day. Determine how
many days it will take Michael to have 80 pieces of sea glass in his collection.
?
d=days
14+3(22)=80
14+3d=80
80=80
d=22
It will take 22 days to have 80 pieces of sea glass.
Try:
a) Kayla is given a monthly allowance. She puts 40% of the money directly into her savings
account. She spends $15 at the movies with her friends. At the end of the month, she has
$3 left. How much is her monthly allowance?
?
a=monthly allowance
0.6(30)-15=3
0.6a-15=3
3=3
a=30
Kayla’s monthly allowance is $30.
b) Juan has already run 6 miles this month. He plans to run an additional 2.5 miles per day.
At this rate, how long will take Juan to run 41 miles?
d=days
6+2.5d=41
d=14
It will take Juan 14 days to run 41 miles.
?
6+2.5(14)=41
41=41
f
© Free to Discover (Amanda Nix) – 2016
Name: _______________________________________ Class: _______ Date: ____________
Solving Two-Step Equations
Practice
For 1-12, solve each two-step equation.
2
d − 9 = −1
3
1)
3a + 2.3 =11.6
2)
4)
4e
+ (−1) = −11
5
1
5) −6 + 4 b =18
2
7)
k −10
= −2
2.5
8)
10)
g
+ 4 = 3.8
25
11)
v
5+m
=0
−3
n + (−2)
5
=1
3)
j
− − (−12) = 0
3
6) 9c + (−4) = −4
9)
5
1 11
− f+ =
2
2 2
12)
h
−1 =1.2
4
© Free to Discover (Amanda Nix) – 2016
For 13-14, use the Five-Step Method.
13) Landon earns about $60 per week mowing lawns in the summer and shoveling driveways in the
winter. He invested $220 in equipment. How long will it take him to earn $860?
14) Mary is thinking of a mystery number. She reduces it by 15% then subtracts 5. The result is 29.
Determine the mystery number.
15) Select the equation that does not belong.
c−2
=0
5
a.
3
x − 7 = −4
2
b.
c.
p 2
+ =1
6 3
d. 4k − 2 = −10
16) Identify and correct error:
3.5x +1.2 =11.7
−1.2 −1.2
3.5 (3.5x ) = (10.5) 3.5
x = 36.75
v
© Free to Discover (Amanda Nix) – 2016
Name: _______________________________________ Class: _______ Date: ____________
Solving Two-Step Equations
Practice
For 1-12, solve each two-step equation.
2
d − 9 = −1
3
1)
3a + 2 =11
2)
4)
4e
−1 = −9
5
5) −6 + 4b =18
7)
k −10
= −2
2
8)
10)
g
+ 4 = −3
2
11)
r
5+m
=0
−3
n + (−2)
5
=1
3)
j
− − (−12) = 0
3
6) 9c + (−4) = −4
9)
5
1 11
− f+ =
2
2 2
12)
h
−1 =1.2
4
© Free to Discover (Amanda Nix) – 2016
For 13-14, use the Five-Step Method.
13) Landon earns about $60 per week mowing lawns in the summer and shoveling driveways in the
winter. He invested $220 in equipment. How long will it take him to earn $860?
14) Meaghan is thinking of a mystery number. She cuts it in half then adds 5. The result is 24.
Determine the mystery number.
15) Select the equation that does not belong.
c−2
=0
5
a.
3
x − 7 = −4
2
b.
c.
p 2
+ =1
6 3
d. 4k − 2 = −10
16) Identify and correct error:
3.5x +1.2 =11.7
−1.2 −1.2
3.5 (3.5x ) = (10.5) 3.5
x = 36.75
r
© Free to Discover (Amanda Nix) – 2016
Name: _______________________________________ Class: _______ Date: ____________
Solving Two-Step Equations
Practice
For 1-9, solve each two-step equation.
2
d − 9 = −1
3
1)
3a + 2 =11
2)
4)
4e
−1 = −9
5
5) −6 + 4b =18
7)
k −10
= −2
2
8)
5+m
=0
−3
3)
j
− − (−12) = 0
3
6) 9c + (−4) = −4
9)
5
1 11
− f+ =
2
2 2
10) Use the Five-Step Method:
Landon earns about $60 per week mowing lawns in the summer and shoveling driveways in the
winter. He invested $220 in equipment. How long will it take him to earn $860?
f
© Free to Discover (Amanda Nix) – 2016
Name: _______________________________________ Class: _______ Date: ____________
2-3
Solving Two-Step Equations
Practice (Answer Key)
For 1-12, solve each two-step equation.
1)
3a + 2.3 =11.6
2)
4e
+ (−1) = −11
5
e=−
7)
b=
8)
10)
g
+ 4 = 3.8
25
g = −5
v
6) 9c + (−4) = −4
16
1
=5
3
3
5+m
=0
−3
c=0
9)
m = −5
k=5
11)
n + (−2)
5
n=7
j
− − (−12) = 0
3
j = 36
1
5) −6 + 4 b =18
2
25
1
= −12
2
2
k −10
= −2
2.5
3)
d =12
a = 3.1
4)
2
d − 9 = −1
3
5
1 11
− f+ =
2
2 2
f = −2
=1
12)
h
−1 =1.2
4
h = 8.8
© Free to Discover (Amanda Nix) – 2016
For 13-14, use the Five-Step Method.
13) Landon earns about $60 per week mowing lawns in the summer and shoveling driveways in the
winter. He invested $220 in equipment. How long will it take him to earn $860?
w=weeks
60w-220=860
w=18
It will take Landon 18 weeks to earn $860.
?
60(18)-220=860
860=860
14) Mary is thinking of a mystery number. She reduces it by 15% then subtracts 5. The result is 29.
Determine the mystery number.
m=mystery number
0.85m-5=29
m=40
The mystery number is 40.
?
0.85(40)-5=29
29=29
15) Select the equation that does not belong.
a.
c.
c−2
=0
5
3
x − 7 = −4
2
b.
p 2
+ =1
6 3
d. 4k − 2 = −10
The solution is -2.
In all others, the
solution is 2.
16) Identify and correct error:
3.5x +1.2 =11.7
−1.2 −1.2
3.5 (3.5x ) = (10.5) 3.5
They should have divided by 3.5.
The correct answer is x=3.
x = 36.75
v
© Free to Discover (Amanda Nix) – 2016
Name: _______________________________________ Class: _______ Date: ____________
2-3
Solving Two-Step Equations
Practice (Answer Key)
For 1-12, solve each two-step equation.
1)
3a + 2 =11
2)
a =3
4)
4e
−1 = −9
5
k −10
= −2
2
g
+ 4 = −3
2
g = −14
r
6) 9c + (−4) = −4
c=0
b =6
8)
5+m
=0
−3
9)
m = −5
11)
n + (−2)
5
n=7
j
− − (−12) = 0
3
j = 36
5) −6 + 4b =18
k=6
10)
3)
d =12
e = −10
7)
2
d − 9 = −1
3
5
1 11
− f+ =
2
2 2
f = −2
=1
12)
h
−1 =1.2
4
h = 8.8
© Free to Discover (Amanda Nix) – 2016
For 13-14, use the Five-Step Method.
13) Landon earns about $60 per week mowing lawns in the summer and shoveling driveways in the
winter. He invested $220 in equipment. How long will it take him to earn $860?
w=weeks
60w-220=860
w=18
It will take Landon 18 weeks to earn $860.
?
60(18)-220=860
860=860
14) Meaghan is thinking of a mystery number. She cuts it in half then adds 5. The result is 24.
Determine the mystery number.
m=mystery number
0.5m+5=24
m=38
The mystery number is 38.
?
0.5(38)+5=24
24=24
15) Select the equation that does not belong.
a.
c.
c−2
=0
5
3
x − 7 = −4
2
b.
p 2
+ =1
6 3
d. 4k − 2 = −10
The solution is -2.
In all others, the
solution is 2.
16) Identify and correct error:
3.5x +1.2 =11.7
−1.2 −1.2
3.5 (3.5x ) = (10.5) 3.5
They should have divided by 3.5.
The correct answer is x=3.
x = 36.75
r
© Free to Discover (Amanda Nix) – 2016
Name: _______________________________________ Class: _______ Date: ____________
2-3
Solving Two-Step Equations
Practice (Answer Key)
For 1-9, solve each two-step equation.
1)
3a + 2 =11
2)
a =3
4)
4e
−1 = −9
5
k −10
= −2
2
k=6
3)
5) −6 + 4b =18
6) 9c + (−4) = −4
c=0
b =6
8)
5+m
=0
−3
m = −5
j
− − (−12) = 0
3
j = 36
d =12
e = −10
7)
2
d − 9 = −1
3
9)
5
1 11
− f+ =
2
2 2
f = −2
10) Use the Five-Step Method:
Landon earns about $60 per week mowing lawns in the summer and shoveling driveways in the
winter. He invested $220 in equipment. How long will it take him to earn $860?
w=weeks
60w-220=860
w=18
It will take Landon 18 weeks to earn $860.
?
60(18)-220=860
860=860
f
© Free to Discover (Amanda Nix) – 2016
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