evaluating algebraic expressions 3-3solving multi-step equations what are the steps for solving...
TRANSCRIPT
Evaluating Algebraic Expressions
3-3 Solving Multi-Step Equations
What are the steps for solving What are the steps for solving multi-step equations?multi-step equations?
How can solving equations be How can solving equations be applied to real life problems?applied to real life problems?
Multi-Step Equations Multi-Step Equations Essential QuestionsEssential Questions
Evaluating Algebraic Expressions
3-3 Solving Multi-Step Equations
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, distributing, or removing the fraction.
Evaluating Algebraic Expressions
3-3 Solving Multi-Step Equations
Solve.
8x + 6 + 3x – 2 = 37
Additional Example 1: Solving Equations That Contain Like Terms
11x + 4 = 37 Combine like terms.
– 4 – 4 Since 4 is added to 11x, subtract 4 from both sides.
11x = 33
x = 3
Since x is multiplied by 11, divide both sides by 11.
3311
11x11
=
8x + 3x + 6 – 2 = 37Commutative Property of Addition
Evaluating Algebraic Expressions
3-3 Solving Multi-Step Equations
Solve.
9x + 5 + 4x – 2 = 42
Check It Out! Example 1
13x + 3 = 42 Combine like terms.
– 3 – 3 Since 3 is added to 13x, subtract 3 from both sides.
13x = 39
x = 3
Since x is multiplied by 13, divide both sides by 13.
3913
13x13
=
9x + 4x + 5 – 2 = 42Commutative Property of Addition
Evaluating Algebraic Expressions
3-3 Solving Multi-Step Equations
Solve 5x + 3(x + 4) = 28.
METHOD 1 Show All Steps METHOD 2 Do Some Steps Mentally
5x + 3(x + 4) = 28
5x + 3x + 12 = 28
8x + 12 = 28
8x + 12 – 12 = 28 – 12
8x = 16
x = 2
8x 168 8
=
5x + 3(x + 4) = 28
5x + 3x + 12 = 28
8x + 12 = 28
8x = 16
x = 2
METHOD 1 Show All Steps
5x + 3(x + 4) = 28
5x + 3x + 12 = 28
8x + 12 = 28
8x + 12 – 12 = 28 – 12
8x = 16
x = 2
8x 168 8
=
Evaluating Algebraic Expressions
3-3 Solving Multi-Step Equations
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.
Evaluating Algebraic Expressions
3-3 Solving Multi-Step Equations
Solve.
+ = –
Additional Example 2A: Solving Equations That Contain Fractions
34
74
5n4
Multiply both sides by 4.74
–3 4
5n4
4 + = 4 ( ) ( )
( ) ( ) ( )5n4
74
–3 44 + 4 = 4
5n + 7 = –3
Distributive Property
( ) ( ) ( )5n4
74
–3 44 + 4 = 4
Simplify.
Evaluating Algebraic Expressions
3-3 Solving Multi-Step EquationsAdditional Example 2A Continued
5n + 7 = –3 – 7 –7 Since 7 is added to 5n, subtract
7 from both sides. 5n = –10
5n5
–10 5
= Since n is multiplied by 5, divide both sides by 5
n = –2
Evaluating Algebraic Expressions
3-3 Solving Multi-Step Equations
The least common denominator (LCD) is the smallest number that each of the denominators will divide into evenly.
Remember!
Evaluating Algebraic Expressions
3-3 Solving Multi-Step Equations
Solve.
+ – =
Additional Example 2B: Solving Equations That Contain Fractions
23
x 2
7x9
17 9
18( ) + 18( ) – 18( ) = 18( )7x9
x2
17 9
23
14x + 9x – 34 = 12
( ) ( )x2
23
7x9
17 918 + – = 18
Distributive Property
Multiply both sides by 18, the LCD.
18( ) + 18( ) – 18( ) = 18( )7x9
x2
17 9
23 Simplify.
2
1 1
9
1
2
1
6
Evaluating Algebraic Expressions
3-3 Solving Multi-Step EquationsAdditional Example 2B Continued
23x = 46
= 23x23
4623 Since x is multiplied by 23, divide
t both sides by 23.x = 2
+ 34 + 34 Since 34 is subtracted from 23x, add 34 to both
sides.
23x – 34 = 12 Combine like terms.
Evaluating Algebraic Expressions
3-3 Solving Multi-Step Equations
Solve.
+ = –
Check It Out! Example 2A
14
54
3n4
Multiply both sides by 4.54
–1 4
3n4
4 + = 4 ( ) ( )
( ) ( ) ( )3n4
54
–1 44 + 4 = 4
3n + 5 = –1
Distributive Property
( ) ( ) ( )3n4
54
–1 44 + 4 = 4
Simplify.
1
1
1
1
1
1
Evaluating Algebraic Expressions
3-3 Solving Multi-Step EquationsCheck It Out! Example 2A Continued
3n + 5 = –1 – 5 –5 Since 5 is added to 3n,
subtract 5 from both sides. 3n = –6
3n3
–6 3
= Since n is multiplied by 3, divide
both sides by 3.n = –2
Evaluating Algebraic Expressions
3-3 Solving Multi-Step Equations
Solve.
+ – =
Check It Out! Example 2B
13
x 3
5x9
13 9
9( ) + 9( ) – 9( ) = 9( )5x9
x3
13 9
13
5x + 3x – 13 = 3
x3
13
5x9
13 9 ( ) ( ) 9 + – = 9
Distributive Property
Multiply both sides by 9, the LCD.
9( ) + 9( ) – 9( ) = 9( )5x9
x3
13 9
13 Simplify.
1
1 1
3
1
1
1
3
Evaluating Algebraic Expressions
3-3 Solving Multi-Step Equations
8x = 16
= 8x8
16 8 Since x is multiplied by 8, divide t
both sides by 8.
x = 2
+ 13 + 13 Since 13 is subtracted from 8x, add 13 to both sides.
8x – 13 = 3 Combine like terms.
Check It Out! Example 2B Continued
Evaluating Algebraic Expressions
3-3 Solving Multi-Step Equations
On Monday, David rides his bicycle m miles in 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.
Additional Example 3: Travel Application
David’s average speed is his total distance for the two days divided by the total time.
average speed
=Total distance
Total time
Evaluating Algebraic Expressions
3-3 Solving Multi-Step EquationsAdditional Example 3 Continued
Multiply both sides by 7.
Substitute m + 3m for total distance and 2 + 5 for total time.2 + 5
= 12 m + 3m
7= 12
4mSimplify.
7 = 7(12) 7
4m
4m = 84
David rode 21.0 miles.
Divide both sides by 4.
m = 21
84 4
4m 4
=
Evaluating Algebraic Expressions
3-3 Solving Multi-Step EquationsCheck It Out! Example 3
Penelope’s average speed is her total distance for the two days divided by the total time.
average speed
=Total distance
Total time
On Saturday, Penelope rode her scooter m 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.
Evaluating Algebraic Expressions
3-3 Solving Multi-Step EquationsCheck It Out! Example 3 Continued
Multiply both sides by 10.
Substitute m + 2m for total distance and 3 + 7 for total time.
3 + 7= 20
m + 2m
10= 20
3mSimplify.
10 = 10(20) 10
3m
3m = 200
Penelope rode approximately 66.7 miles.
Divide both sides by 3.
m 66.67
200 3
3m 3
=