LESSON 8.2 Equations Expressions, - Mustang...
Transcript of LESSON 8.2 Equations Expressions, - Mustang...
How do you solve a two-step equation??
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ESSENTIAL QUESTION
L E S S O N
8.2Solving Two-Step Equations
Modeling and Solving Two-Step Equations You can solve two-step equations using algebra tiles.
Use algebra tiles to model and solve 3n + 2 = 11.
Model the equation.
Remove 2 +1-tiles from
each side of the mat.
Divide each side into
3 equal groups.
The solution is n = 3.
EXAMPLEXAMPLE 1
STEP 1
STEP 2
STEP 3
STEP 4
Use algebra tiles to model and solve each equation.
YOUR TURN
1. 2x + 5 = 11 2. 3n - 1 = 8
3. 2a - 3 = -5 4. -4y + 2 = -2
Expressions, equations, and relationships—7.10.B Represent solutions for one-variable, two-step equations and inequalities on number lines. Also 7.11.A, 7.11.B
7.11.A
Since there are +1-tiles on both sides of the equation, you can remove, or subtract, 2 +1-tiles from each side to help isolate the variable.
x = 3
a = -1
n = 3
y = 1
251Lesson 8.2
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Reflect5. Analyze Relationships Describe how you could find the weight of one
baseball bat using only arithmetic. Compare your method with the one
used in Example 2.
Representing Solutions on a Number LineYou have used inverse operations to solve equations with one operation. You
can use the same method to solve equations with more than one operation.
After solving, you can represent the solution on a number line.
Tony carried 5 identical baseball bats to a ball game inside a carrying
case weighing 12 ounces. The combined weight of the bats and the case
was 162 ounces. How much did each bat weigh? Graph the solution on
a number line.
Write an equation to represent the problem.
Let w = the weight of a bat in ounces.
5 times the weight of each bat plus 12 oz is 162 oz.
5w + 12 = 162
Use inverse operations to solve the equation.
5w + 12 = 162
- 12 _____ - 12 _____
5w = 150
5w ___ 5
= 150 ____ 5
w = 30
Each bat weighed 30 ounces
Graph the solution on a number line.
EXAMPLE 2
STEP 1
STEP 2
STEP 3
Math TalkMathematical Processes
7.10.B
Could you solve the equation in Example 2 by first dividing both sides
by 5? Explain.
Subtract 12 from both sides.
Divide both sides by 5.
It is helpful to reverse the order of operations when solving equations that have more than one operation.
Unit 4252
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Write and solve an equation that represents the situation. Graph the
solution on the number line.
6. Maureen wants to buy a $198
camera. She has $30 and plans
to save $12 each week. In how
many weeks will she be able
to buy the camera?
7. A rectangular picture
frame has a perimeter of
58 inches. The height of the
frame is 18 inches. What is the
width of the frame?
YOUR TURN
Determining if a Given Value Makes an Equation True You can use substitution to decide whether a given value is the solution
of an equation.
After first doubling the weight being pulled by a dog sled, the sled driver
removes 20 pounds. The final weight of the dog sled is 180 pounds. The
equation 2w - 20 = 180 can be used to find w, the initial weight of the
sled. Determine which, if any, of these values is a solution: w = 60;
w = 80; w = 100.
Substitute each value for w in the equation 2w - 20 = 180.
w = 60 w = 80 w = 100
2(60) - 20 = 180 2(80) - 20 = 180 2(100) - 20 = 180
Evaluate to see if a true equation results.
2(60) - 20 ? =
180 2(80) - 20
? =
180 2(100) - 20
? =
180
120 - 20 ? =
180 160 - 20
? =
180 200 - 20
? =
180
100 ? =
180 140
? =
180 180
? =
180
not true not true true
The initial weight of the sled was 100 pounds.
EXAMPLEXAMPLE 3
STEP 1
STEP 2
✗✗ ✔
7.11.B
253Lesson 8.2
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Guided Practice
The equation 2x + 1 = 9 is modeled below. (Example 1)
1. To solve the equation with algebra tiles, first remove .
Then divide each side into .
2. The solution is x = .
3. 8m - 15 = 41
m =
4.
k =
k _ 3 + 21 = 27
5. 9p - 18 = 27
p = 3; p = 5; p = 7
6. a = -10; a = 0; a = 10
a ___ -2
- 5 = 0
Determine which, if any, of the given values is a solution. (Example 3)
Solve each equation. Then graph the solution on the number line. (Example 2)
Determine which, if any, of the given values is a solution.
8. 3k + 15 = 66
k = -7; k = 17; k = 27
9. p
_ 9
- 5 = 7
p = -72; p = 18; p = 108
YOUR TURN
7. How can you decide which operations to use to solve a two-step
equation?
ESSENTIAL QUESTION CHECK-IN??
Unit 4254
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Name Class Date
Independent Practice
Solve.
8. 9s + 3 = 57 9. 4d + 6 = 42 10. -3y + 12 = -48
11.
12. 13.
14. -9h - 15 = 93
15. 16.
17.
18. 19. 46 = -6t -8
k _ 2
+ 9 = 30g
__ 3
- 7 = 15 z _ 5
+ 3 = -35
24 + n __ 4
= 10 -17 + b __ 8
= 13
-5 = 9 + c _ 4
-3 + p
__ 7
= -5
8.2
27. -5.5x + 0.56 = -1.64 28. -4.2x + 31.5 = -65.1 29. k ___ 5.2
+ 81.9 = 47.2
20. After making a deposit, Puja had $264 in her savings account.
She noticed that if she added $26 to the amount originally in the
account and doubled the sum, she would get the new amount.
How much did she originally have in the account?
21. The current temperature in Smalltown is 20 °F. This is 6 degrees less
than twice the temperature that it was six hours ago. What was the
temperature in Smalltown six hours ago?
22. Daphne gave away 3 more than half of her apples. She gave away
17 apples in all. How many apples did Daphne have originally?
23. Artaud noticed that if he takes the opposite of his age and adds 40
he gets the number 28. How old is Artaud?
24. Sven has 11 more than twice as many customers as when he
started selling newspapers. He now has 73 customers. How many
did he have when he started?
25. Paula bought a ski jacket on sale for $6 less than half its original
price. She paid $88 for the jacket. What was the original price?
26. Michelle has a starting balance on a gift card for $300. She buys
several dresses at $40 a piece. After her purchases she has $140 left
on the gift card. How many dresses did she buy?
Use a calculator to solve each equation.
7.10.B, 7.11.A, 7.11.B
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Work Area30. Write a two-step equation involving multiplication and subtraction that
has a solution of x = 7.
31. Write a two-step equation involving division and addition that has
a solution of x = -25
32. Reason Abstractly The formula F = 1.8C + 32 allows you to find the
Fahrenheit (F) temperature for a given Celsius (C) temperature. Solve the
equation for C to produce a formula for finding the Celsius temperature
for a given Fahrenheit temperature.
33. Reason Abstractly The equation P = 2(ℓ + w) can be used to find the
perimeter P of a rectangle with length ℓ and width w. Solve the equation
for w to produce a formula for finding the width of a rectangle given its
perimeter and length.
34. Critique Reasoning A student’s solution to the equation 3x + 2 = 15
is shown. Describe the error that the student made.
35. Multiple Representations Explain how you could use the work
backward problem-solving strategy to solve the equation x _ 4 − 6 = 2.
36. Reason Abstractly Solve the equation ax + b = c for x.
FOCUS ON HIGHER ORDER THINKING
3x + 2 = 15
x + 2 = 5
x = 3
Divide both sides by 3.
Subtract 2 from both sides.
Unit 4256
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