Download Solving Multi - Step Equations

Solving Multi-Step Equations
Lesson 2.3
0011 0010 1010 1101 0001 0100 1011
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2
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California
0011 0010 1010 1101 0001 0100 1011
Standards
4.0 Students simplify expressions before
solving linear equations and inequalities in one
variable, such as 3(2x – 5) + 4(x – 2) = 12.
5.0 Students solve multistep problems,
including word problems, involving linear
equations and linear inequalities in one variable and
provide justification for each step.
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2
4
Solving Multi-step Equation
• Steps –
0011 0010
1010 1101
0001
0100 1011
1. Circle
like
terms.
2. Combine like terms.
3. Isolate Variable
1. Add/subtract
2. Multiply/divide
3. Solve
2x + 3x – 4 =11
5x – 4 = 11
+4 +4
1
5x = 15
5
5
x=3
2
4
More Example
Solve the equation. Check your answer.
0011 0010 1010 1101 0001 0100 1011
1.undo the division.
2. undo the addition.
2x + 1 = 21
–1 –1
2x
= 20
x = 10
3. undo the multiplication.
4. The solution set is {10}.
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2
4
Now let check our answer
Solve
the1010
equation.
0011
0010
1101Check
0001 your
0100answer.
1011
Check
To check your solution, substitute 10
for x in the original equation.
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7 7

2
4
Let Try again
Solve 4 = 2x + 5 – 6x
4 = 2x + 5 – 6x
0011 0010 1010 1101 0001 0100 1011
4 = 2x – 6x + 5
4 = –4x + 5
–5
–5
–1 = –4x
Combine like terms.
undo the addition.
undo the multiplication.
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2
4
The solution set is
.
Try some on your own.
1. 46 = 4x – 4 + 6x
4.
0011 0010 1010 1101 0001 0100 1011
5=x
2. 4x – 8 + 2x = 40
x= 8
3. 36 = 10a – 12 – 7a
16 = a
5.
1
x=6
6. –8 – 2d + 2 = 4
2
4
d = -5
You may have to combine like terms
or use the Distributive Property
before you begin solving.
0011 0010 1010 1101 0001 0100 1011
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4
Using Distributive Property
Solve the equation.
0011 0010 1010 1101 0001 0100 1011
5(p – 2) = –15
5(p – 2) = –15
5(p) + 5(–2) = –15
5p – 10 = –15
+10 +10
5p
= –5
Distribute 5.
Simplify.
1
2
Since 10 is subtracted from 5p, add 10 to
both sides.
4
Since p is multiplied by 5, divide both sides by
5.
p = –1
The solution set is {–1}.
0011 0010 1010 1101 0001 0100 1011
Helpful Hint
You can think of a negative sign as a coefficient
of –1.
–(x + 2) = –1(x + 2) and –x = –1x.
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Another Example
Solve the equation.
0011 0010 1010 1101 0001 0100 1011
10y – (4y + 8) = –20
10y +(–1)(4y + 8) = –20
Write subtraction as the addition of
the opposite.
10y + (–1)(4y) + (–1)(8) = –20 Distribute –1.
10y – 4y – 8 = –20
6y – 8 = –20
+8
6y
+8
= –12
y= -2
Simplify.
1
2
4
Combine like terms.
Since 8 is subtracted from 6y, add 8
to both sides to undo the
subtraction.
Try some on your own.
1. –4(2 – y) = 8 y = 4
4. d + 3(d – 4) = 20
0011 0010 1010 1101 0001 0100 1011
2. 3(x + 1) – 4 = 5
d=8
X=2
3. x – (12 – x) = 38
25
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2
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Application
Lin sold 4 more shirts than Greg. Fran sold 3 times as
0011 0010 1010 1101 0001 0100 1011
many shirts as Lin. In total, the three sold 51 shirts.
How many shirts did Greg sell?
Locate key words in the question. UNDERLINE WHAT
YOU ARE BEING ASKED TO FIND
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2
Reread and circle relevant information
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2
4
Lin sold 4 more shirts than Greg. Fran sold 3 times as
0011 0010 1010 1101 0001 0100 1011
many shirts as Lin. In total, the three sold 51 shirts.
How many shirts did Greg sell?
3
Reread the part of the problem you underlined, and
define an appropriate variable (or variables)
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2
Since the information is given in relation
to Lin, set an equation for each individual
in terms of Lin.
Greg = L – 4
Lin = L
Fran = 3L
4
Lin sold 4 more shirts than Greg. Fran sold 3 times as
many shirts as Lin. In total, the three sold 51 shirts.
0011 0010
1101
0001 did
0100Greg
1011 sell?
How1010
many
shirts
Write an equation (or inequality) and then check to
see if it’s correct by rereading the circled and
(L – 4) + (L) + underlined
(3L) = 51information
Substitute.
–4=
51 Combine like Greg
Greg + Lin5L
+ Fran
= 51
terms. = L – 4
+4 +4
Lin = L
Since 4 is subtracted from 5L add 4 to
5L
= 55
= the
3Lsubtraction.
both sidesFran
to undo
4
1
L = 11
2
4
Since L is multiplied by 5, divide both
sides by 5 to undo the
multiplication.
More Application
At a local gym, there is a joining fee of
0011 0010
1010 1101
0001a0100
1011
$59.95
and
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?
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Step
1:
Step 3:
4:
2:
2
Let
m represent
monthly
feefee
paidisby each.
initial
total
12 theplus
Monthly
cost.
for 2
months
fee for 2
2
(12m
+
119.90)
4
=
1319.90
2(12m + 59.95) = 1319.90
0011
0010 1010+1101
0001 0100=1011
2(12m)
2(59.95)
1319.90 Distribute 2.
24m + 119.90 = 1319.90
–119.90 –119.90 Since 119.90 is added to 24m,
subtract 119.90 from both
24m
= 1200.00 sides to undo the addition.
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2
Since m is multiplied by 24, divide
both sides by 24 to undo the
multiplication.
4
m = 50
Sara and Martin each paid $50 per month.
Lesson Quiz: Part l
Solve
each
0011
0010 1010
1101equation.
0001 0100 1011
1. 2y + 29 – 8y = 5
2. 3(x – 9) = 30
19
3. x – (12 – x) = 38
4.
4
9
25
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2
4
5. If 3b – (6 – b) = –22, find the value of 7b.–28
Lesson Quiz: Part ll
0011 0010 1010 1101 0001 0100 1011
6. Josie bought 4 cases of sports drinks for an
upcoming meet. After talking to her coach,
she bought 3 more cases and spent an
additional $6.95 on other items. Her receipts
totaled $74.15. Write and solve an equation
to find how much each case of sports drinks
cost.
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4
4c + 3c + 6.95 = 74.15; $9.60