3.2 Solving Equations Using Addition and Subtraction
Objectives:
* To solve linear equations using addition and subtraction
* To use linear equations to solve word problems involving real-world situations
Solving Linear Equations Using Addition
In this strategy, we balance the sides of the equation as we solve for the variable.
To solve equations like , you can use mental math. Ask yourself, “What number minus 3 is equivalent to 6?”
This strategy can work for easier problems, but we need a better plan so we can solve more difficult problems.
First, we must “undo” the minus 3. The inverse
(opposite) of subtracting 3 is _________________ 3.
But we have to be fair! If we are going to add 3 to one side of the equation, we MUST add 3 to the other side to BALANCE the equation.
3
+ = 6 + 3 3
Solving Linear Equations Using Addition
Examples:
1) 2)
3) 4)
We must isolate the variable. What number is on the same side of the = with the variable? How do you undo this operation?
Solving Linear Equations Using Subtraction
First, we must “undo” the plus 2. The inverse (opposite) of adding 2 is ______________ 2.
But we have to be fair! If we are going to subtract 2 from one side of the equation, we MUST
subtract 2 from the other side to BALANCE the equation.
The process for solving equations using subtraction is very similar to solving equations using addition.
How do you undo addition?
Example:
Simplifying First
What needs to be done before you can solve these??
1) 2)
Negative Signs with Variables
In some problems, you will see a negative sign in front of the variable you are solving for.
Example: What does this
mean?
The negative sign means “the opposite of x,” meaning the opposite sign, positive or negative.
To solve, we change the sign of the other side of the equation.
-x = -5
x = -12
Several record temperature changes have taken place in Spearfish, South Dakota. On January 22, 1943, the temperature in Spearfish fell
from 54 degrees Fahrenheit at 9:00 am to -4 degrees Fahrenheit at 9:27 am. By how many degrees did the temperature fall?
You started with some money in your pocket. All you spent was $4.65 on lunch. You ended up with $7.39 in your pocket. Write an equation
to find out how much money you started with.
Word Problems- SWEET!
Can you solve one – step problems involving addition and subtraction? Assignment: pg 135 : A: 3, 21-44all, 68-74 even B: 33-41, 45-53, 68, 70
Even-Numbered Answers for 3.1 Assignment
20) Subtract 2 ½ 34) b = 4 52) 22) x =5 36) b = - 8 68) 4x + 8 24) r = 15 38) a = 3 70) 5y + 35 26) n = - 4 40) a =- 1 72) –x2 + 6x 28) a = 3/5 42) C; 22 CDs 74) 16y – 14y2 30) y = 0 44) A; 46) score = 7291 32) x = - 11 48) 50) x = 5☺ Warm – Up ☺
1) 2x – x(3x – 5) =
2) (6x – 3)(-4x)=
3)
3.2 Solving Equations Using
Multiplication
To solve equations like , you will need to use multiplication.
In this problem, the x is being divided by 4. To solve for x, we will need to do the inverse of dividing by 4, which is multiplying by 4.
**Don’t forget that you will need to do this to BOTH sides of the equation to keep it balanced!
Ex)
Objectives: To solve linear equations using multiplication and division and to use linear equations to solve word problems involving real-world situations
Solving Equations Using
Division
To solve equations like , we can use division.
In this problem, the x is being multiplied by 4. To solve for x, we will need to do the inverse of multiplying by 4, which is DIVIDING by 4.
**Don’t forget that you will need to do this to BOTH sides of the equation to keep it balanced!
Let’s try it!
Remember – locate the variable and
undo whatever operation is being done to it!
Fraction Time
☺
How do we solve an equation like this?
Right now, the x is being multiplied by . The inverse of multiplying by is dividing by .
Ex)
How do we divide by fractions? Multiply by the RECIPROCAL!
Ex) It takes 6 cans of chili to make one batch of George’s extra-special chili-cheese dip. How many cans does it take to make 3 and a half
batches? Write an equation and solve it. Be sure to state what your variables represent.
Ex) Amy’s mom has been passing out cookies to the members of Amy’s soccer team. If each of the team members (including 15 players, 2
coaches, and a manager) receive 3 cookies with no extra cookies left, how many cookies did Amy’s mom bring? Write an equation an solve it.
Can you solve equations involving multiplication and division? Assignment: pg 142: A or B
3.2 Solutions
14. Multiply by 6 36. 246 16. Multiply by -4 38. 0 18. Multiply by 40. 30 20. No 42. 24 22. Yes 44. 17 24. 11 46. 16 26. -9 50. 400 students 28. 6 52. 540 peanuts 30. -8 54. 45 sec 32. 56. 43.2 by 64.8 feet 58. 3 feet 34. -10 60. B; 1.5 tspWarm-Up
You will be asked to solve problems like which require more than one step to solve.
Guidelines for Solving Multi-Step Equations:
1) Simplify both sides if necessary distribute & combine like terms 2) To solve you must undo the order of operations BACKWARDS!
Start with Undo add or subtract
Then Undo multiplication and division
There is nothing to simplify What goes
first?
Simplify to get a new equation.
What goes next?
3.3 Multi-Step Equations
Objectives: To use 2 or more steps to solve a linear equation and to use multi-step equations to solve word problems
How can you check you answer? Do I expect you to show work on assignments, quizzes and tests?
Examples:
Remember Do you need to
simplify before you start.
1) 2)
simplify distribute
Combine like terms
Lots of steps!
Where should you
start??
Multiplying by a Reciprocal First
Given the example , what would you do first?
Let’s try something that may make this simpler!! Undo the first.
Ex) The bill (parts and labor) for the repair of a car was $400. The cost of parts was $125, and the labor cost was $50 per hour. Write and solve and equation to find the number of hours of labor.
(HINT: this would make a GREAT quiz question!!! ☺)
Can you solve multi-step equations? How do you know what operation to undo first?
3.3a Solutions
16.
4 32.
-52
52.
14 months
18.
12 34.
15
56.
5 hours
20.
70 36.
66.
35, 37, 39
22.
-12
38.
-1
no
24.
-16
40.
5
26.
-3 42.
left side was
28.
3
simplified incorrectly
30.
50b. 27, 28, 29
Quiz Review Warm Up
1) -2(3x – 10) = 20 2) 3x – 5(x +8) = 28
2) 3) 5x – 18 = 29 4)
5) The sum of 2 consecutive even numbers is 122. Variable:
Equation:
6) Fred charges $25 for delivery and $30 per hour for installation . If the bill was $ 155, how long did Fred work?
Variable: Equation:
3.3b Worksheet
Solutions
3.4 Solving Equations with Variables on
Both Sides
Objective: To solve equations with variables on both sides of the equation.
Warm-up:
How would you rate yourself on solving these problems?
GREAT!! OK – getting there Need some help, but ok Need to
1st: Variables to one side How do you decide who to move?
2nd: Constants to the other side Who must you move?
7x + 19 = -2x + 55
1) Which side has the smallercoefficient?
2) Add 2x to both sides. 3) Simplify.
9x + 19 = 55
4) Subtract 19 from both sides.
9x = 36
6) Divide both sides by 9. 5) Simplify.
+2x +2x
- 19 -19
9 9
7) Simplify.
Can you check you
answer? How?
Does Mrs. W want to see
these steps? DUH!
1) 80 – 9y = 6y
Let’s try some!!
3) 4(1 - x) + 3x = -2(x + 1)
How is this one different?
More Examples…You try these!!
2 More of the fun type!!
Can you solve equations with
variables on both sides of the
equal sign?
Assignment: Pg 157 A or B
7) 2(3 – 2x) + x = -3(x + 1) 8) 3(x – 5) – 6 = -21 + 3x
3.4a Solutions
10.
B
32.
All Real Numbers
18.
-27
34.
5
20.
36.
-2
38.
.5
22.
No Solution
40.
All Real Numbers
24.
No Solution
42.
The distributive property
26.
1
was applied improperly.
28.
-1
44.
True for all Real Numbers3.4.b Solutions
1. 9 2. -6 3. No Solution 4. 2 5. All Reals 6. 2
7. -17 8. 7 9. -52/2
10. No Solution 11. 1 12. All Reals 13. ½ 14. -12 15. 6
16. 3 17. 4 18. 23/3
19. 61/8 20. -11 21. 4
3.6 Solving Decimal Equations
Objective: To find exact and approximate solutions of equations that contain decimals.
Solve the equation. Round your result to the nearest hundredth.
9)Jenny and some of her friends are going to see a movie. Jenny has generously offered to pay for all of the tickets and the snacks. The cost of a student’s ticket is $6.25, and their group spent a total of $48.36 on snacks. If the total amount of money spent is $85.86, how many movie tickets did Jenny buy?
10) Max goes to the deli and buys some ham and some turkey. Ham costs $4.29 per pound and turkey costs $3.89 per pound. If Max purchased 1.5 pounds of ham, and his total bill was $15.19, how much turkey did he buy? Round your answer to the nearest hundredth.
3.6 Solutions
14. - 35.2, -35.19 38. 456.2
16. -3.0, -2.97 44. answers will vary 18. 0.1, 0.11 46. answers will vary
24. -1.58 50. C; about 200 people 26. 9.62 28. 0.31 30. 1.56 32. 0.38 34. -6.98 36. -1.33
3.6 Solutions
14. - 35.2, -35.19 38. 456.2
16. -3.0, -2.97 44. answers will vary
18. 0.1, 0.11 46. answers will vary 24. -1.58 50. C; about 200 people 26. 9.62 28. 0.31 30. 1.56 32. 0.38 34. -6.98 36. -1.33
