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Solving Multi-Step Equations
Section 2-3 Part 1
Goals
Goal
• To solve multi-step
equations in one variable.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• None
Solving Multi-Step Equations
Equations that are more complicated may have to be
simplified before they can be solved. You may have
to combine like terms or use the Distributive
Property before you begin using inverse operations.
Solving Multi-Step Equations
A martial arts school is offering a special where new students can
enroll for half price, after a $12.50 application fee.
Ten students enrolled and paid a total of $325. To find the regular
price of enrollment, you can solve an equation.
Regular price of enrollment
Number of
students
Total
cost
Application fee
Solving Multi-Step Equations
Notice that this equation contains multiplication,
division, and addition. An equation that contains
multiple operations will require multiple steps to solve.
You will create an equivalent equation at each step.
Example: Solving Multi-Step
Equations
Solve the equation. Check your answer.
Since 2x + 1 is divided by 3, multiply both
sides by 3 to undo the division.
2x + 1 = 21
–1 –1
2x
= 20
Since 1 is added to 2x, subtract 1 from
both sides to undo the addition.
Since x is multiplied by 2, divide both
sides by 2 to undo the multiplication.
x = 10
The solution set is {10}.
Example: Continued
Check
To check your solution,
substitute 10 for x in the
original equation.
7
7
Example: Solving Multi-Step
Equations
Solve the equation.
Since 3x – 4 is divided by 2, multiply both
sides by 2 to undo the division.
+4
18 = 3x
+4
Since 4 is subtracted from 3x, add 4 to
both sides to undo the subtraction.
Since x is multiplied by 3, divide both
sides by 3 to undo the multiplication.
6 = x or x = 6
The solution set is {6}.
Your Turn:
Solve the equation.
Since 5m + 13 is divided by 2, multiply
both sides by 2 to undo the division.
5m + 13 = 2
–13 –13
5m
= –11
Since 13 is added to 5m, subtract 13
from both sides to undo the addition.
Since m is multiplied by 5, divide both
sides by 5 to undo the multiplication.
The solution set is
.
Your Turn:
Solve the equation.
Since 4 – 2x is divided by 4, multiply both
sides by 4 to undo the division.
4 – 2x = –8
–4
–4
–2x = –12
x=6
Since 4 is added to – 2x, subtract 4 from
both sides to undo the addition.
Since x is multiplied by –2, divide both
sides by –2 to undo the
multiplication.
The solution set is {6}.
Solving Multi-Step Equations
You may have to combine like terms or use the
Distributive Property before you begin solving.
Like terms
Constant
4x – 3x + 2
Example: Combining Like
Terms and Solving Equations
Solve 8x – 21 – 5x = –15
8x – 21 – 5x = –15
8x – 5x – 21 = –15
3x – 21 = –15
+21 = +21
3x
= 6
Use the Commutative Property of
Addition. Combine like terms.
Since 21 is subtracted from 3x, add 21 to
both sides to undo the subtraction.
Since x is multiplied by 3, divide both
sides by 3 to undo the multiplication.
x=2
The solution set is {2}.
Example: Combining Like
Terms and Solving Equations
Solve 4 = 2x + 5 – 6x
4 = 2x + 5 – 6x
4 = 2x – 6x + 5
4 = –4x + 5
–5
–5
–1 = –4x
Use the Commutative Property of
Addition. Combine like terms.
Since 5 is added to –4x, subtract 5 from
both sides to undo the addition.
Since x is multiplied by –4, divide both
sides by –4 to undo the
multiplication.
The solution set is
Your Turn:
Solve the equation.
2a + 3 – 8a = 8
2a – 8a +3 = 8
–6a + 3 = 8
–3 –3
–6a
=5
Use the Commutative Property of
Addition. Combine like terms.
Since 3 is added to –6a, subtract 3 from
both sides to undo the addition.
Since a is multiplied by –6, divide both
sides by –6 to undo the
multiplication.
The solution set is
.
Your Turn:
Solve the equation.
–8 – 2d + 2 = 4
–8 – 2d + 2 = 4
–2d + 2 – 8 = 4
–2d –6 = 4
+6 +6
–2d
Use the Commutative Property of
Addition. Combine like terms.
Since 6 is subtracted from –2d, add 6 to
both sides to undo the subtraction.
= 10
Since d is multiplied by –2, divide both
sides by –2 to undo the
multiplication.
d = –5
The solution set is {–5}.
Your Turn:
Solve the equation.
4x – 8 + 2x = 40
4x – 8 + 2x = 40
4x + 2x – 8 = 40
6x – 8 = 40
+8 +8
6x
Use the Commutative Property of
Addition. Combine like terms.
Since 8 is subtracted from 6x, add 8 to
both sides to undo the subtraction.
= 48
Since x is multiplied by 6, divide both
sides by 6 to undo the multiplication.
x=8
The solution set is {8}.
Example: Distributive Property
and Solving Equations
Solve the equation.
5(p – 2) = –15
5(p – 2) = –15
5(p) + 5(–2) = –15
5p – 10 = –15
+10 +10
5p
= –5
Distribute 5.
Simplify.
Since 10 is subtracted from 5p, add 10 to
both sides.
Since p is multiplied by 5, divide both
sides by 5.
p = –1
The solution set is {–1}.
Helpful Hint
You can think of a negative sign as a coefficient of –1.
–(x + 2) = –1(x + 2) and –x = –1x.
Example: Distributive Property
and Solving Equations
Solve the equation.
10y – (4y + 8) = –20
10y +(–1)(4y + 8) = –20
10y + (–1)(4y) + (–1)(8) = –20
10y – 4y – 8 = –20
6y – 8 = –20
+8 +8
6y
= –12
Write subtraction as the addition
of the opposite.
Distribute –1.
Simplify.
Combine like terms.
Since 8 is subtracted from 6y,
add 8 to both sides to undo
the subtraction.
Example: Continued
6y = –12
Since y is multiplied by 6, divide both
sides by 6 to undo the multiplication.
y = –2
Your Turn;
Solve the equation.
3(a + 1) – 4 = 5
3(a + 1) – 4 = 5
(3)(a) + (3)(1) – 4 = 5
3a + 3 – 4 = 5
3a – 1 = 5
+ 1 +1
3a
= 6
Distribute 3.
Simplify. Combine like terms.
Since 1 is subtracted from 3a, add 1 to
both sides to undo the subtraction.
Since a is multiplied by 3, divide both
sides by 3 to undo the multiplication.
a=2
Your Turn:
Solve the equation.
–4(2 – y) = 8
–4(2 – y) = 8
(–4)(2) + (–4)(–y) = 8
–8 + 4y = 8
+8
+8
4y = 16
Distribute –4 .
Simplify.
Since –8 is added to 4y, add 8 to both
sides.
Since y is multiplied by 4, divide both
sides by 4 to undo the multiplication.
y=4
Your Turn:
Solve the equation.
d + 3(d – 4) = 20
d + 3(d – 4) = 20
d + 3(d) + 3(–4) = 20
d + 3d – 12 = 20
4d – 12 = 20
+12 +12
4d
= 32
Distribute 3.
Simplify.
Combine like terms.
Since 12 is subtracted from 4d, add 12 to
both sides to undo the subtraction.
Since d is multiplied by 4, divide both
sides by 4 to undo the multiplication.
d=8
Multiplying by a Reciprocal First
6 (x + 3)
Solve 66 = –
5
6
= – systematically
(x + 3)
Solving 66
equations
5
is an example of deductive
6 (x + 3)
reasoning.
– 5 66 =Notice
– 5how
– each
6
6
5
solution step is based on number
properties or properties of
–55 = x + 3
equality.
–58 = x
It is easier to solve this equation
if you don’t distribute – 6 first.
5
Write original equation.
6
Multiply by reciprocal of – 5 .
Simplify.
Subtract 3 from each side.
Example: Application
Lin sold 4 more shirts than Greg. Fran sold 3
times as many shirts as Lin. In total, the three
sold 51 shirts. How many shirts did Greg sell?
To determine the number of shirts sold write an
equation: G + L + F = 51.
Since the information is given in relation to Lin, set an
equation for each individual in terms of Lin.
G=L–4
F = 3L
L=L
Example: Continued
Lin sold 4 more shirts than Greg. Fran sold 3
times as many shirts as Lin. In total, the three
sold 51 shirts. How many shirts did Greg sell?
G + L + F = 51
Substitute.
(L – 4) + (L) + (3L) = 51
Combine like terms.
5L – 4 = 51
+4 +4
Since 4 is subtracted from 5L add 4
to both sides to undo the
5L
= 55
subtraction.
L = 11
Since L is multiplied by 5, divide
both sides by 5 to undo the
multiplication.
Example: Continued
Lin sold 4 more shirts than Greg. Fran sold 3
times as many shirts as Lin. In total, the three
sold 51 shirts. How many shirts did Greg sell?
G=L–4
= 11 – 4
=7
Greg sold 7 shirts.
Your Turn:
At a local gym, there is a joining fee of $59.95 and a monthly
membership fee. Sara and Martin both joined this gym.
Their combined cost for 12 months was $1319.90. How much
is the monthly fee?
Let m represent the monthly fee paid by each.
Monthly fee
for 2
2
12 months
plus
(12m
+
initial fee for
2
119.90)
is
total
cost.
=
1319.90
Your Turn: Continued
2(12m + 59.95) = 1319.90
2(12m) + 2(59.95) = 1319.90
24m + 119.90 = 1319.90
–119.90 –119.90
24m
= 1200.00
Distribute 2.
Since 119.90 is added to 24m,
subtract 119.90 from both
sides to undo the addition.
Since m is multiplied by 24, divide
both sides by 24 to undo the
multiplication.
m = 50
Sara and Martin each paid $50 per month.
Your Turn:
Lily and 4 of her friends want to enroll in a yoga class.
After enrollment, the studio requires a $7 processing fee.
The 5 girls pay a total of $125.75. How much does the class
cost?
Let c represent the cost of the class.
number
enrolled
5
class cost
(c
plus
processing fee
+
7)
is
=
total cost
125.75
Your Turn: Continued
5(c + 7) = 125.75
5(c) + 5(7) = 125.75
5c + 35 = 125.75
– 35 – 35
5c
Distribute 5.
Since 35 is added to 5c, subtract
35 from both sides to undo the
addition.
= 90.75
Since c is multiplied by 5, divide
both sides by 5 to undo the
multiplication.
c = 18.15
The cost per person is $18.15 a month.
Joke Time
• How did the Vikings send secret messages?
• By norse code!
• What was Camelot?
• A place where people parked their camels!
• Why did the teacher put the lights on?
• Because the class was so dim!
Assignment
• 2.3 Pt 1 Exercises Pg. 103 – 104: #8 – 28
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