3-3 Solving Multi-Step Equations
Preview
Evaluating Algebraic Expressions
Warm Up
California Standards
Lesson Presentation
3-3 Solving Multi-Step Equations
Warm
Up
Evaluating
Algebraic Expressions
Solve.
1. 3x = 102
x = 34
2. y = 15 y = 225
15
3. z – 100 = 21 z = 121
4. 1.1 + 5w = 98.6 w = 19.5
3-3 Solving Multi-Step Equations
California
Evaluating
Algebraic Expressions
Standards
Extension of
AF4.1 Solve two-step
linear equations and inequalities in one variable
over the rational numbers, interpret the solution or
solutions in the context from which they arose, and
verify the reasonableness of the results.
3-3 Solving Multi-Step Equations
Evaluating Algebraic Expressions
A multi-step equation requires more
than two steps to solve. To solve a
multi-step equation, you may have to
simplify the equation first by combining
like terms.
3-3 Solving Multi-Step Equations
Additional Example 1: Solving Equations That
Contain Like Terms
Evaluating
Solve.
Algebraic Expressions
8x + 6 + 3x – 2 = 37
8x + 3x + 6 – 2 = 37
11x + 4 = 37
–4
11x
–4
= 33
11x = 33
11 11
x=3
Commutative Property of
Addition
Combine like terms.
Since 4 is added to 11x,
subtract 4 from both sides.
Since x is multiplied by 11,
divide both sides by 11.
3-3 Solving Multi-Step Equations
Check It Out! Example 1
Solve.
Evaluating
Algebraic Expressions
9x + 5 + 4x – 2 = 42
9x + 4x + 5 – 2 = 42
13x + 3 = 42
–3
13x
–3
= 39
13x = 39
13 13
x=3
Commutative Property of
Addition
Combine like terms.
Since 3 is added to 13x,
subtract 3 from both sides.
Since x is multiplied by 13,
divide both sides by 13.
3-3 Solving Multi-Step Equations
Evaluating Algebraic Expressions
If an equation contains fractions, it may
help to multiply both sides of the
equation by the least common
denominator (LCD) to clear the fractions
before you isolate the variable.
3-3 Solving Multi-Step Equations
Additional Example 2A: Solving Equations That
Contain Fractions
Evaluating Algebraic Expressions
Solve.
5n + 7= – 3
4
4
4
(
) ( )
7 = 4 –3
4(5n
+
4
(4 ) (4)
4)
7 = 4 –3
4(5n
+
4
(4 ) (4)
4)
4 5n + 7
4
4
= 4 –3
4
5n + 7 = –3
Multiply both sides by 4.
Distributive Property
Simplify.
3-3 Solving Multi-Step Equations
Additional Example 2A Continued
Evaluating Algebraic Expressions
5n + 7 = –3
– 7 –7 Since 7 is added to 5n, subtract
7 from both sides.
5n
= –10
5n= –10
5
5
n = –2
Since n is multiplied by 5, divide both
sides by 5
3-3 Solving Multi-Step Equations
Evaluating Algebraic Expressions
Remember!
The least common denominator (LCD) is the
smallest number that each of the denominators
will divide into evenly.
3-3 Solving Multi-Step Equations
Additional Example 2B: Solving Equations That
Contain Fractions
Evaluating Algebraic Expressions
Solve.
7x + x – 17 = 2
3
2
9
9
Multiply both
17
18 7x + x – 9 = 18 2 sides by 18, the
9
2
3
LCD.
(
()
()
2 Distributive
= 18 3 Property
()
()
2
= 18 3
()
()
17
– 18 9
()
()
17
– 18 9
7x
x
18 9 + 18 2
9
7x
x
18 9 + 18 2
2
)
1
1
2
1
6
( ) Simplify.
14x + 9x – 34 = 12
1
3-3 Solving Multi-Step Equations
Additional Example 2B Continued
23x –Evaluating
34 = 12
Combine
like terms.
Algebraic
Expressions
+ 34 + 34
23x
= 46
sides.
23x = 46
23 23
x=2
Since 34 is subtracted from
23x, add 34 to both
t
Since x is multiplied by 23, divide
both sides by 23.
3-3 Solving Multi-Step Equations
Check It Out! Example 2A
Solve.
Evaluating
3n+ 5 = – 1
4
4
4
Algebraic Expressions
(
) ( )
5 = 4 –1
4(3n
+
4
(4 ) (4)
4)
5 = 4 –1
4(3n
+
4
(4 ) (4)
4)
4 3n + 5
4
4
1
= 4 –1
4
1
1
1
1
3n + 5 = –1
1
Multiply both sides by 4.
Distributive Property
Simplify.
3-3 Solving Multi-Step Equations
Check It Out! Example 2A Continued
Evaluating Algebraic Expressions
3n + 5 = –1
– 5 –5
3n
= –6
3n= –6
3
3
n = –2
Since 5 is added to 3n,
subtract 5 from both sides.
Since n is multiplied by 3, divide
both sides by 3.
3-3 Solving Multi-Step Equations
Check It Out! Example 2B
Solve.
Evaluating Algebraic
5x + x – 13 = 1
3
3
9
9
(
)
Expressions
()
Multiply both
sides by 9, the
LCD.
()
Distributive
Property
x – 13 = 9 1
9 5x
+
3 9
9
3
()
5x
9 9 + 9
()
()
x
3
() ()
3
5x
x
9 9 +9 3
1
1
()
13
– 9 9
1
1
= 9 3
3
13
–9 9 =9
1
1
()
5x + 3x – 13 = 3
1
3
1
Simplify.
3-3 Solving Multi-Step Equations
Check It Out! Example 2B Continued
8x –Evaluating
13 = 3
Combine
like terms.
Algebraic
Expressions
+ 13 + 13
8x
= 16
8x = 16
8
8
x=2
Since 13 is subtracted from 8x,
add 13 to both sides.
t
Since x is multiplied by 8, divide
both sides by 8.
3-3 Solving Multi-Step Equations
Additional Example 3: Travel Application
On Monday,
DavidAlgebraic
rides his bicycle
m miles in
Evaluating
Expressions
2 hours. On Tuesday, he rides three times as
far in 5 hours. If his average speed for the two
days is 12 mi/h, how far did he ride on
Monday? Round your answer to the nearest
tenth of a mile.
David’s average speed is his total distance for the
two days divided by the total time.
Total distance
Total time
= average speed
3-3 Solving Multi-Step Equations
Additional Example 3 Continued
m + 3m
= 12
Evaluating
2+5
4m
= 12
7
Substitute m + 3m for total
Algebraic
Expressions
distance and 2
+ 5 for total time.
Simplify.
7 4m = 7(12) Multiply both sides by 7.
7
4m = 84
4m = 84
Divide both sides by 4.
4
4
m = 21
David rode 21.0 miles.
3-3 Solving Multi-Step Equations
Check It Out! Example 3
On Saturday, Penelope rode her scooter m
Evaluating
Algebraic
Expressions
miles
in 3 hours. On
Sunday, she
rides twice
as far in 7 hours. If her average speed for two
days is 20 mi/h, how far did she ride on
Saturday? Round your answer to the nearest
tenth of a mile.
Penelope’s average speed is her total distance for
the two days divided by the total time.
Total distance
Total time
= average speed
3-3 Solving Multi-Step Equations
Check It Out! Example 3 Continued
m + 2m
= 20
Evaluating
3+7
3m
= 20
10
Substitute m + 2m for total
Algebraic
distance and Expressions
3 + 7 for total
time.
Simplify.
10 3m = 10(20) Multiply both sides by 10.
10
3m = 200
3m = 200 Divide both sides by 3.
3
3
m  66.67
Penelope rode approximately 66.7 miles.
3-3 Solving Multi-Step Equations
Lesson Quiz
Solve.
Evaluating Algebraic Expressions
1. 6x + 3x – x + 9 = 33 x = 3
2. 29 = 5x + 21 + 3x
3.
5 + x = 33
8
8
8
4.
6x–
7
2x =
21
x=1
x = 28
25
21
9
x = 116
5. Linda is paid double her normal hourly rate for each
hour she works over 40 hours in a week. Last week
she worked 52 hours and earned $544. What is her
hourly rate? $8.50