Transcript 2-3 Solving Multi Step Equations

2-3 Solving Multi-Step Equations
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Warm Up
California Standards
Lesson Presentation
2-3 Solving Multi-Step Equations
Warm Up
Evaluate each expression.
1.
–4
2. 26 – 4(7 – 5) 18
Simplify each expression.
5. 10c + c 11c
6. 8.2b + 3.8b – 12b 0
7. 5m + 2(2m – 7) 9m – 14
8. 6x – (2x + 5)
4x – 5
2-3 Solving Multi-Step Equations
California
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.
2-3 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
2-3 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.
2-3 Solving Multi-Step Equations
Additional Example 1A: Solving Two-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
x = 10
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.
The solution set is {10}.
2-3 Solving Multi-Step Equations
Additional Example 1A Continued
Solve the equation. Check your answer.
Check
To check your solution,
substitute 10 for x in the
original equation.
7
7

2-3 Solving Multi-Step Equations
Additional Example 1B: Solving Two-Step Equations
Solve the equation. Check your answer.
Since 3x – 4 is divided by 2,
multiply both sides by 2 to undo
the division.
+4
+4
18 = 3x
6=x
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.
The solution set is {6}.
2-3 Solving Multi-Step Equations
Additional Example 1B Continued
Solve the equation. Check your answer.
Check
To check your solution,
substitute 6 for x in the
original equation.
7
7
2-3 Solving Multi-Step Equations
Check It Out! Example 1a
Solve the equation. Check your answer.
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
.
2-3 Solving Multi-Step Equations
Check It Out! Example 1a Continued
Solve the equation. Check your answer.
Check
To check your solution,
substitute
for m in the
original equation.
1
1
2-3 Solving Multi-Step Equations
Check It Out! Example 1b
Solve the equation. Check your answer.
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}.
2-3 Solving Multi-Step Equations
Check It Out! Example 1b Continued
Solve the equation. Check your answer.
Check
To check your solution,
substitute 6 for x in
the original equation.
–2
–2 
2-3 Solving Multi-Step Equations
You may have to combine like terms or
use the Distributive Property before you
begin solving.
2-3 Solving Multi-Step Equations
Additional Example 2A: Simplifying Before Solving
Equations
Solve 8x – 21 – 5x = –15
8x – 21 – 5x = –15
8x – 5x – 21 = –15
3x – 21 = –15
+21 = +21
3x
= 6
x=2
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.
The solution set is {2}.
2-3 Solving Multi-Step Equations
Additional Example 2B: Simplifying Before Solving
Equations
Solve 4 = 2x + 5 – 6x
4 = 2x + 5 – 6x
Use the Commutative Property of
4 = 2x – 6x + 5
Addition. Combine like terms.
4 = –4x + 5
Since 5 is added to –4x, subtract
–5
–5
5 from both sides to undo the
–1 = –4x
addition.
Since x is multiplied by –4, divide
both sides by –4 to undo the
multiplication.
The solution set is
.
2-3 Solving Multi-Step Equations
Check It Out! Example 2a
Solve the equation. Check your answer.
2a + 3 – 8a = 8
2a – 8a +3 = 8
Use the Commutative Property of
Addition. Combine like terms.
–6a + 3 = 8
–3 –3
–6a = 5
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
.
2-3 Solving Multi-Step Equations
Check It Out! Example 2a Continued
Solve the equation. Check your answer.
Check 2a + 3 – 8a = 8
To check your solution,
substitute
for a in
the original equation.
8
8

2-3 Solving Multi-Step Equations
Check It Out! Example 2b
Solve the equation. Check your answer.
–8 – 2d + 2 = 4
–8 – 2d + 2 = 4
–2d + 2 – 8 = 4
Use the Commutative Property of
Addition. Combine like terms.
–2d –6 = 4
+6 +6
–2d
= 10
Since 6 is subtracted from –2d,
add 6 to both sides to undo the
subtraction.
Since d is multiplied by –2, divide
both sides by –2 to undo the
multiplication.
d = –5
The solution set is {–5}.
2-3 Solving Multi-Step Equations
Check It Out! Example 2b Continued
Solve the equation. Check your answer.
Check
–8 – 2d + 2 = 4
–8 – 2(–5) + 2
4
To check your solution,
–8 + 10 + 2
2+2
4
4
substitute –5 for d in
4
4
the original equation.
2-3 Solving Multi-Step Equations
Check It Out! Example 2c
Solve the equation. Check your answer.
4x – 8 + 2x = 40
Use the Commutative Property of
4x – 8 + 2x = 40
Addition. Combine like terms.
4x + 2x – 8 = 40
6x – 8 = 40
+8 +8
6x
= 48
Since 8 is subtracted from 6x, add
8 to both sides to undo the
subtraction.
Since x is multiplied by 6, divide
both sides by 6 to undo the
multiplication.
x=8
The solution set is {8}.
2-3 Solving Multi-Step Equations
Check It Out! Example 2c Continued
Solve the equation. Check your answer.
Check
4x – 8 + 2x = 40
4(8) – 8 + 2(8) 40
32 – 8 + 16
24 + 16
40
40
40
40 
To check your solution,
substitute 8 for x in
the original equation.
2-3 Solving Multi-Step Equations
Additional Example 3A: Simplify Using the
Distributive Property
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}.
2-3 Solving Multi-Step Equations
Helpful Hint
You can think of a negative sign as a coefficient
of –1.
–(x + 2) = –1(x + 2) and –x = –1x.
2-3 Solving Multi-Step Equations
Additional Example 3B: Simplify Using the
Distributive Property
Solve the equation.
10y – (4y + 8) = –20
Write subtraction as the
addition of the opposite.
10y +(–1)(4y + 8) = –20
10y + (–1)(4y) + (–1)(8) = –20 Distribute –1.
10y – 4y – 8 = –20 Simplify.
6y – 8 = –20 Combine like terms.
Since 8 is subtracted from
+8
+8
6y, add 8 to both sides
to undo the subtraction.
6y
= –12
2-3 Solving Multi-Step Equations
Additional Example 3B Continued
Solve the equation.
10y – (4y +8) = –20
6y = –12
y = –2
Since y is multiplied by 6, divide
both sides by 6 to undo the
multiplication.
2-3 Solving Multi-Step Equations
Check It Out! Example 3a
Solve the equation. Check your answer.
3(a + 1) – 4 = 5
3(a + 1) – 4 = 5 Distribute 3.
(3)(a) + (3)(1) – 4 = 5
3a + 3 – 4 = 5 Simplify. Combine like terms.
3a – 1 = 5 Since 1 is subtracted from 3a, add
+ 1 +1
1 to both sides to undo the
3a
= 6
subtraction.
Since a is multiplied by 3, divide
both sides by 3 to undo the
multiplication.
a=2
2-3 Solving Multi-Step Equations
Check It Out! Example 3a Continued
Solve the equation. Check your answer.
Check
3(a + 1) – 4 = 5
3(2 + 1) – 4
3(3) – 4
5
5
To check your solution,
9–4
5
the original equation.
5
5
substitute 2 for a in

2-3 Solving Multi-Step Equations
Check It Out! Example 3b
Solve the equation. Check your answer.
–4(2 – y) = 8
–4(2 – y) = 8
(–4)(2) + (–4)(–y) = 8
Distribute –4 .
Simplify.
–8 + 4y = 8 Since –8 is added to 4y, add 8 to
+8
+8
both sides.
4y = 16
Since y is multiplied by 4, divide
both sides by 4 to undo the
multiplication.
y=4
2-3 Solving Multi-Step Equations
Check It Out! Example 3b Continued
Solve the equation. Check your answer.
Check
–4(2 – y) = 8
–4(2 – 4) 8
–4(–2) 8
8
8
To check your solution,
substitute 4 for y in
the original equation.
2-3 Solving Multi-Step Equations
Check It Out! Example 3c
Solve the equation. Check your answer.
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
d=8
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.
2-3 Solving Multi-Step Equations
Check It Out! Example 3c Continued
Solve the equation. Check your answer.
Check d + 3(d – 4) = 20
8 + 3(8 – 4)
20
To check your solution,
8 + 3(4)
20
substitute 8 for d in
20
the original equation.
20

2-3 Solving Multi-Step Equations
Additional Example 4: 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
2-3 Solving Multi-Step Equations
Additional Example 4 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
5L
= 55
add 4 to both sides to
undo the subtraction.
L = 11
Since L is multiplied by 5,
divide both sides by 5 to
undo the multiplication.
2-3 Solving Multi-Step Equations
Additional Example 4 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.
2-3 Solving Multi-Step Equations
Check It Out! Example 4a
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
plus initial fee
for 2
months
(12m
+
119.90)
is
total
cost.
=
1319.90
2-3 Solving Multi-Step Equations
Check It Out! Example 4a Continued
2(12m + 59.95) = 1319.90
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
= 1200.00 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.
2-3 Solving Multi-Step Equations
Check It Out! Example 4b
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
is
total
cost
+
7)
=
125.75
2-3 Solving Multi-Step Equations
Check It Out! Example 4b Continued
5(c + 7) = 125.75
5(c) + 5(7) = 125.75
5c + 35 = 125.75
– 35 – 35
5c = 90.75
c = 18.15
Distribute 5.
Since 35 is added to 5c,
subtract 35 from both
sides to undo the addition.
Since c is multiplied by 5,
divide both sides by 5 to
undo the multiplication.
The cost per person is $18.15 a month.
2-3 Solving Multi-Step Equations
Lesson Quiz: Part l
Solve each equation.
1. 2y + 29 – 8y = 5
4
2. 3(x – 9) = 30 19
3. x – (12 – x) = 38 25
4.
9
5. If 3b – (6 – b) = –22, find the value of 7b. –28
2-3 Solving Multi-Step Equations
Lesson Quiz: Part ll
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.
4c + 3c + 6.95 = 74.15; $9.60