1. Solving Linear Equations
Chapter 3
Solving Linear Equations
3.1 Solving Equations Using Addition & Subtraction
Objectives:
1. To solve linear equations using addition and subtraction
2. To use linear equations to solve word problems involving real-world situations

2. Solving Linear Equations Using Addition
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.
In this strategy, we balance the sides of the equation as we solve for the variable.
Since the “6” is the quantity after x subtract “3”. So, first, we must “undo” the minus 3. The inverse (opposite) of subtracting 3 is _________________ 3.
add 3 + 6 = 3 x -
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.
Don’t forget to check your answer!!!

3. Solving Linear Equations Using Addition
We must isolate the variable. What number is on the same side of the = with the variable? How do you undo this operation?
Examples: 1) 2) 4 + 55 + 3) 4) 6 + 7 +
How can you check your answers?

4. Solving Linear Equations Using Subtraction
The process for solving equations using subtraction is very similar to solving equations using addition.
How do you undo addition?
Example:
First, we must “undo” the plus 2. The inverse (opposite) of adding 2 is ______________ 2.
subtract
- 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. 7 – 15 – 3 –

5. Simplifying First What needs to be done before you can solve these??1) 2) 3) 4) 2 – 2 + 4 – 5 +

6. Negative Signs with Variables
In some problems, you will see a negative sign in front of the variable you are solving for.
Another way is that you imagine you are the variable and the negative sign is the image.
What does this mean?
Example:
x = -12
The negative sign means “the opposite of x,” meaning the opposite sign, positive or negative.
Thinking of that you are standing in front of a mirror (the zero), where is your image? Or the reverse question: if you know where is your image, where are you standing?
Think the variable “x” is yourself and then the “–x” is your image in the mirror. 1 2 3 4 5 6 7 8 9 10 11 -3 -2 -1 -4 -5 -6 -7 -8 -9
-10

7. Negative Signs with Variables
To solve, we change the sign of the other side of the equation.
–x = –5

8. Word Problems - SWEET! – 54 + 4.65
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. 54 –
4.65 +

9. Summary
Inverse operations are operations that undo each other, such as “+” and “–”, “ · ” and “”.
You must apply the inverse operation at both sides of the equation at the same time. So the equation is kept in balance.
When the variable has a negative sign, you can imagine you are the variable and the negative sign represents the image.

10. 3.2 Solving Equations Using Multiplication
Objectives: To solve linear equations using multiplication and division and to use linear equations to solve word problems involving real-world situations
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)
Don’t forget to check your answer!

11. 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!
Ex) –6 –6

12. You try it! Remember – locate the variable and undo whatever operation is being done to it! Also, simplify before you start!

13. 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 .
How do we divide by fractions?
Multiply by the RECIPROCAL!

14. 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 .
How do we divide by fractions?
Multiply by the RECIPROCAL!
Ex)

15. Can you solve equations involving multiplication and division?
Ex) Amy’s mom has made 72 cookies to the members of Amy’s soccer team. If each of the team members (including 15 players, 2 coaches, and a manager) receives same number of cookies with no extra cookies left, how many cookies did Amy’s team member receive? Write an equation an solve it.
Suppose each team member received n cookies, then
18 n = 72
n = 4
Ex) It takes 16 cans of chili to make 4 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.
Suppose we need x cans of chili to make 1 batch of chili-cheese dip, then
4 x = 16
(3.5) · x = (3.5) · 4 = 14
x = 4
Can you solve equations involving multiplication and division?

16. Summary
Always keep BOTH sides of the equation balanced when solving equation using multiplication and division.
If the number in front of the variable is a whole number, simply divide this number at BOTH sides of the equation.
If the number in front of the variable is a fraction, it is better to multiply the reciprocal of the this fraction at BOTH sides of the equation.

17. 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
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?
How can you check you answer? You MUST show all work on assignments, quizzes and tests.

18. Examples: Remember  Do you need to simplify before you start.2) 1) +2 –5 3) 4) +8 –6

19. Lots of steps! Where should you start??
Distribute
simplify
Combine like terms

20. Lots of steps! Where should you start??
Subtract 3
Simplify
Combine constant like terms
Multiply by the reciprocal of – 3/4 –3

21. 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. HOW? 2 11 +7 –3

22. Ex) The bill (parts and labor) for the repair of a car was $400
Ex) The bill (parts and labor) for the repair of a car was $400. The cost of parts was $150, 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!!! )
Let the number of hours of labor be x, then the labor cost is 50x and the parts cost is $150. The total cost $400 is the parts cost and labor cost. So
50 x = 400
– –150
50 x = 250
x = 5

23. Summary
When solving a multiple step equation, simplify both sides before you start. You may use distributive property, combine like terms, or multiply a fraction.
To solve you must undo the order of operations BACKWARDS! (inverse operation)
Start with  Undo add or subtract (+ or – )
Then  Undo multiplication and division ( · or  )

24. 3.4 Solving Equations with Variables on Both Sides
Objective: To solve equations with variables on both sides of the equation.
Warm-up:
6x – 2 = –10
2x – 6x + 20 = –16
6x = –8
–4x + 20 = –16
x = –4/3
–4x + 20 = –16 –5
–20
–4x = –36
x = 9 +5
How would you rate yourself on solving these problems?
GREAT!! OK – getting there Need some help, but ok Need to come in for help

25. Can you check you answer? How?
1st: Variables to one side  How do you decide who to move? 2nd: Constants to the other side  Who must you move?
1) Which side has the smaller coefficient?
7x + 19 = -2x – 17
+2x x
2) Add 2x to both sides.
9x + 19 = –17
3) Simplify.
–19 –19
4) Subtract 19 from both sides.
9x = –36
5) Simplify.
6) Divide both sides by 9.
7) Simplify.
Can you check you answer? How?

26. Can you check you answer? How?
1st: Variables to one side  How do you decide who to move? 2nd: Constants to the other side  Who must you move?
1) Which side has the smaller coefficient?
6x + 22 = -3x + 31
+3x x
2) Add 3x to both sides.
9x + 22 = 31
3) Simplify.
–22 –22
4) Subtract 22 from both sides.
9x = 9
5) Simplify.
6) Divide both sides by 9.
7) Simplify.
Can you check you answer? How?

27. How is this one different?
Let’s try some!!
1) 80 – 9y = 6y
2) 64 – 12 w = 6w
+9y
+12w
3) 4(1 – x) + 3x = –2(x + 1)
How is this one different?
4 – 4x + 3x = –2x – 2
3x + 4 = 10 – 3x + 6
4 – x = –2x – 2
3x + 4 = 16 – 3x
+2x +2x
+3x x
4 + x = – 2
6x + 4 = 16 –4 –4
x = – 6
x = 2
6x = 12

28. 1st: Distribution  How do you decide? Why?
Ex.
4(1 – x) + 3x = –2(x + 1)
1) Distribution.
4 – 4x + 3x = –2x – 2
2) Simplify.
–x + 4 = –2x – 2
+2x x
3) Add 2x to both sides.
x + 4 = –2
4) Simplify.
–4 –4
5) Subtract 4 from both sides.
x = –6
6) Simplify.
Can you check you answer? How?

29. +3x +3x –4 –4 6 6 1st: Distribution  How do you decide? Why?
2) Simplify.
+3x x
3) Add 3x to both sides.
4) Simplify.
– –4
5) Subtract 4 from both sides.
6) Simplify.
7) Divide 6.
8) Simplify.

30. More Examples…You try these!!
5) 9x + 22 = –3x + 46
+3x x
4x + 6 = 18 – 4x + 12
12x + 22 = 46
4x + 6 = 30 – 4x
–22 –22
+4x x
12x = 24
8x + 6 = 30
–6 –6
8x = 24

31. 2 More of the fun type!! +3x +3x +3x +3x 6 – 4x + x = –3x – 3
–21 = –21
identity
6 = – 3
inconsistent
This kind of equation is called “identity equation”. When you get an identity equation, you just write
and conclude that
Many solutions
This kind of equation is called “inconsistent equation”. When you get an inconsistent equation, you just write
and conclude that
No solution

32. 2 More questions of the fun type!!
9) -4(1+2x) - 2x = –5(2x+1)
10) 2(3x – 1) – 6 = x
– 4 – 8x – 2x = –10x – 5
6x – 2 – 6 = –8 + 6x
6x – 8 = –8 + 6x
–4 – 10x = –10x – 5
–6x –6x
+10x x
–8 = –8
identity
–4 = – 5
inconsistent
This kind of equation is called “identity equation”. When you get an identity equation, you just write
and conclude that
Many solutions
This kind of equation is called “inconsistent equation”. When you get an inconsistent equation, you just write
and conclude that
No solution

33. Summary
When solving the equations with variables at both sides, you need apply the distributive property, combining like terms, and other skills.
Always simplify the equation first.
Always work on the variable before to work on the constant. Move the variables to one side and the constants to the other side.
Moving the variable term with less coefficient to the other side that the variable term with larger coefficient.
Be aware of two extreme situations:
No solution  you meet an inconsistent equation when solving.
Many solutions  you meet an identity equation when solving.

34. Assignment
P 157 #’s

35. 3.5 Linear Equations and Problem Solving Word Problems!!! My Favorite
Keys to succeed!
Draw a picture
Write down important information
Define your variable!!
Put the info in a chart if you can

36. We will meet the following typical types of real-life application questions:
Consecutive Integers
Geometry
Traveling
Tickets
Accounting
The real-life application questions are not just limited to the those above types. Others may be:
6. Clock (combination of Geometry and Traveling)
7. Work
8. Mixture

37. Consecutive Integers
1) Find three consecutive integers whose sum is 162.
Integer 1
x - 1
Integer 2 x
Integer 3
x + 1
Total
162

38. Consecutive Integers x - 2 x
You Try This!
2) The measures of the angles of a certain triangle are consecutive even integers. Find their measures.
Angle 1
x - 2
Angle 2 x
Angle 3
x + 2
Total
180 1 2 3

39. Geometry
3) A board is 12 ft long and is to be cut into 3 pieces so that the second piece is twice the size of the first piece, and the third is three times the size of the second piece. Find the length of the 3 pieces of board. 1 2 3
Analysis: second = 2 · first, third = 3 · second = 3 · (2 · first) = 6 · first
Piece 1 x
Piece 2 2x
Piece 3 6x

40. You Try This!
Geometry
4) The longest side of a triangle is 3 inches more than twice the middle side. The shortest side is 2 inches less than the middle side. If the perimeter is 45 inches, how long is each side?
Longest
2x+3
Middle x
Shortest
x – 2
Perimeter 45
2x+3
x-2 x

41. Traveling
5) A pair of hikers, 18 miles apart, begin at the same time to hike toward each other. If one walks at a rate that is 1 mph faster than the other, and if they meet 2 hours later, how fast is each one walking?
Hiker 1’s dist.
Hiker 2’s dist. 18
Hiker 1’s distance + Hiker 2’s distance = 18
Hiker 1 x 2 2x
Hiker 2
x+1
2(x+1)
Rate
Time
Distance = 18

42. Traveling
You Try This!
6) A pair of cars, 280 miles apart, begin at the same time to run toward each other. If car A from city A runs at a rate that is 10 mph faster than car B from city B, and if they meet 2 hours later, how far is the place they meet away from city A?
Car A’s dist.
Car B’s dist.
280 A B
Car A’s distance + Car B’s distance = 280
Car A
x + 10 2
2(x + 10)
Car B x 2x
Rate
Time
Distance = 280

43. Traveling
7) The Yankee Clipper leaves the pier at 9:00am at 8 knots (nautical miles per hour). A half hour later, The Riverboat Rover leaves the same pier in the same direction traveling at 10 knots. At what time will The Riverboat Rover overtake The Yankee Clipper?
Yankee Clipper
9:00 ~ 9:30
Traveled
4 nt. miles 8
x hours after 9:30 8x
8x + 4
Riverboat Rover
0 nt. miles 10
10x
0 + 10x
rate
time
dist.
total
Yankee Total = Riverboat Rover Total
4 nt. mi.
8x nt. mi. YC YC YC RR
10x nt. mi. RR
x hr. after
9:30
9:00
9:30

44. Tickets 2 x 2x 4 450 – x 4(450 – x) ----- 450 1160
8) The school play sold 450 tickets for a total of $ If student tickets are $2.00 and adult tickets are $4.00, how many of each type were sold?
Student 2 x 2x
Adult 4
450 – x
4(450 – x)
Total
-----
450
1160
Student tickets sales + Adult tickets sales = 1160

45. Tickets 3 x 3x 5 72 – x 5(72 – x) 72 258 You Try This!
9) Fred is selling tickets for his home movies. Tickets for friends are $3.00 and everyone else must pay $5.00 per ticket. If he sold 72 tickets and made $258 how many of each type did he sell?
Friend 3 x 3x
Non-Friend 5
72 – x
5(72 – x)
Total
---- 72
258

46. Accounting
10) Barney has $450 and spends $3 each week. Betty has $120 and saves $8 each week. How many weeks will it take for them to have the same amount of money?
Barney
450 3 x
450 – 3x
Betty
120 8 x
initial
wk spend wk
end total

47. Accounting 100 4 x 100 + 4x 28 10 28 + 10x initial wk sp wk end total
You Try This!
11) Fred has $100 and saves $4 each week. Wilma has $28 and saves $10 each week. How long will it take for them to have the same amount of money? What is that amount?
Fred
100 4 x x
Wilma 28 10 x
initial
wk sp wk
end total

48. More on Consecutive Integers
12) Find three consecutive integers that the difference of the product of two larger ones and the product of two smaller ones is 30.
Integer 1
x - 1
Integer 2 x
Integer 3
x + 1
Prod. of Larger 2
x(x + 1)
Prod. of Smaller 2
x(x - 1)

49. More on Traveling
13) A driver averaged 50mph on the highway and 30mph on the side roads. If the trip of 185 miles took a total of 4 hours and 30 minutes, how many miles were on the highway.
Highway 50 x
x/50
Side Road 30
185 – x
(185 – x)/30
Total
185
4.5
My God! It is so complicated!!!

50. More on Traveling 50 x 50x 30 4.5 – x 30(4.5 – x) 4.5 185
13) A driver averaged 50mph on the highway and 30mph on the side roads. If the trip of 185 miles took a total of 4 hours and 30 minutes, how many miles were on the highway.
Highway 50 x
50x
Side Road 30
4.5 – x
30(4.5 – x)
Total
4.5
185

51. Weighted Averages
14) You have 32 coins made up of dimes and nickels. You have a total of $ How many of each type of coin do you have?
Dime 10 x
10x
Nickel 5
32 – x
5(32 – x)
Total 32
285

52. Weighted Averages
15) The Quick Mart has two kinds of nuts. Pecans sell for $1.55 per pound and walnuts sell for $1.95 per pound. How many pounds of walnuts must be added to 15 pounds of pecans to make a mixture that sells for $1.75 per pound.
Pecans
1.55 15
15 · 1.55
Walnuts
1.95 x
1.95x
Mixture
1.75
x +15
1.75(x + 15)

53. Mixture
16) A druggist must make 20 oz of a 12% saline solution from his supply of 5% and 15% solutions. How much of each should he use?
12% solution
12% 20
20·12%
5% solution 5% x
x · 5%
15% solution
15%
20 – x
(20 – x) ·15%

54. 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. 1) 2)

55. Solve the equation. Round your results to the nearest thousandth.

56. Solve the equation. Round your results to the nearest thousandth.

57. 9)Jenny and some of her friends are going to see a movie
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.

58. Summary Assignment P. 169 #’ 14 - 38
Solving the decimal equation is the same as solving the any other equation we have learned.
Assignment
P. 169 #’

59. 3.7 Formulas and Functions
Objective:
Solve a formula for one of its variable
Rewrite an equation in function form
Formula -- an algebraic equation that relates two or more real-life quantities.
Example 1 Solving and using an area formula
Solve A = lw for w. (In other words, isolate w.)
b) Find the width of a rectangle that has an area of 42 ft2 and a length of 6 ft.

60. Example 2 Solving and using an area formula
Solve I = Prt for t.
b) Find the number of years t that $2800 was invested to earn $504 at 4.5%.

61. Example 3 Solve: V = r2h for h
r2 r2 V
r2
= h
Example 4 Solve: 3x + y = 4 for y
-3x x
y = –3x + 4

62. Practice 1 Solve: A = LW for L
W W A W
= L
Practice 2 Solve: I = P r t for P
I = P r t
r t r t I rt
P =

63. Example 5 Solve: –3x – 5 = 13
+5 +5
1 type inv. op.
– 3x = 18
3.2 question
– 3x = 18
– –3
1 type inv. op.
x = 6
Example 6 Solve: mx – a = k for x
+a +a
1 type inv. op.
mx = k + a
3.2 question
mx = k + a
m m
1 type inv. op.
k + a m
x =

64. Example 7 Solve for y:
–8x –8x
y = –12x +

65. Practice Solve for y
4x + y = –3x – 4y = –10

66. -2x -2x -1 -1 Rewriting an equation in Function Form
A two-variable equation is written in function form if one of its variables is isolated on one side of the equation. The isolated variable is the output and is a function of the input. For instance, the equation P = 4s describes the perimeter P of a square as a function of its side length s.
Example 8 Rewrite the equation 2x – y = 9, so that y is a function of x.
-2x x

67. Example 9
Write the equation 2x – y = 9, so that x is a function of y.
b) Use the result to find x when y = –2, –1, 0 and 1
+ y + y y x -2
(–2 – 9)/2 = –5.5 -1
(–1 – 9)/2 = –5
(0– 9)/2 = –4.5 1
(1– 9)/2 = –4

68. Example 10 Solve for y: 4(y + 2) = 8 – 3x
4 4
–8 –8
y + 2 =
4y = – 3x
–2 –2
4y = –3x
y =
y =

69. Example Solve for F.

70. Example 12 Solve for y:
–2x –2x
–2x –2x
y + 3 = 10x
–3 –3
y = 10x – 3
y = 10x – 3

71. Example 13 Solve for w: 4 4 4 1 4
w + 12x = –8x+12
–12x – 12x
w = –20x+12

72. Practice Solve for w 2.

73. Example 14 The Pathfinder was launched on December 4, 1996
Example The Pathfinder was launched on December 4, During its 212-day slight to Mars, it traveled about 310 miles. Estimate the time the Pathfinder would have taken to reach Mars traveling at the following speeds.
30,000 miles per hour
40,000 miles per hour
60,000 miles per hour
80,000 miles per hour
We must use the formula d = r t and solve for the time t.

74. Example 14 The Pathfinder was launched on December 4, 1996
Example The Pathfinder was launched on December 4, During its 212-day slight to Mars, it traveled about 310 million miles. Estimate the time the Pathfinder would have taken to reach Mars traveling at the following speeds.
30,000 miles per hour
40,000 miles per hour
60,000 miles per hour
80,000 miles per hour
We must use the formula d = r t and solve for the time t.

75. Example 14 The Pathfinder was launched on December 4, 1996
Example The Pathfinder was launched on December 4, During its 212-day slight to Mars, it traveled about 310 miles. Estimate the time the Pathfinder would have taken to reach Mars traveling at the following speeds.
30,000 miles per hour
40,000 miles per hour
60,000 miles per hour
80,000 miles per hour
We must use the formula d = r t and solve for the time t.

76. We must use the formula d = r t and solve for the time t.
Example The Pathfinder was launched on December 4, During its 212-day slight to Mars, it traveled about 310 miles. Estimate the time the Pathfinder would have taken to reach Mars traveling at the following speeds.
30,000 miles per hour
40,000 miles per hour
60,000 miles per hour
80,000 miles per hour
We must use the formula d = r t and solve for the time t.
Which of the four speeds is the best estimate of Pathfinder’s average speed?

77. Summary
Solving a formula for one of its variable is the same as solving an equation.
Rewriting an equation in function form is the same as solving a formula for one of its variable.