This document provides definitions and examples for solving linear equations in one and two variables. It begins by defining key terms like equation, linear equation, independent variable, dependent variable, identity, and conditional. Examples are given to classify equations as identities or conditionals. The document then demonstrates solving linear equations algebraically by using properties of equality and inverse operations. Tables and graphs are presented as additional methods for solving equations visually. The document concludes by introducing solving equations with variables on both sides and solving linear inequalities.

1. 3.1
Linear Equations
In 1 and 2 Variables

2. Use your book glossary to find the definitions
1. EQUATION:
2. LINEAR EQUATION: an equation with exponent of 1
(in 1 variable)
(in 2 variables)
3. INDEPENDENT VARIABLE:
4. DEPENDENT VARIABLE:
A statement of equality between 2 quantities

3. 5. IDENTITY: One side of equal sign is the same as the
other. All values of the variable produce a true statement.
6. CONDITIONAL: Equation that is true for only certain
values of the variable.

4. For the following, decide whether the equation is an
identity or conditional and write the answer on the line.
a. x + 2 = 7
(can x be anything? Or just one thing?)
b. x – 3 = y
(can x be anything? Can y be anything? Or just one thing?)
conditional
conditional

5. c. ab = ba
d. P = 2l + 2w
e. 2(x + 4) = 2x + 8
f. 4r – 1r = 3r
Which are equations
in 1 variable?
Which are equations
in 2 variables?
identity
conditional
identity
identity

6. Translate the following into equations.
Let x be the number.
a. One fourth of a number is 24.
b. A number multiplied by 3 and then added to 6 gets 15.
c. The difference between fourteen and twice a number is
negative 8.
(1/4)x = 24
6 + 3x = 15
14 – 2x = -8

7. For the following, translate the algebraic
equation into an English sentence. Can you
think of another way to write the same thing?
x – 2 = -4
9x = 72
2 less than a number is negative four.
Nine times a number is 72.

8. For the following, define the variables and find the
equation
a. The cost of renting a car at $35 flat fee and $0.11 per mile.
b. The total cost of an item bought in Michigan; including tax.
x = number of miles y = cost or renting a car
y = 35 + 0.11x
x = ticket cost of an item y = total cost of item
y = x + 0.06x

9. The next set of sentences contains a phrase that
describes a need for grouping symbols. Underline the
phrase and then write the equation.
a. Five times the difference between nine and a number is –35.
b. The four times the sum of a number and 5 is 24.
c. The total cost of an air-compressor is $5 for the first day
plus $30 for each additional day.
5(9 – x) = -35
4(x + 5) = 24
y = 5 + 30(x – 1)
* x= number of days

10. Birthday Party Exploration
Emma wants to have her birthday party at Valley Bowling
Alley. Her mother can rent a party room, let the kids rent
shoes and play 3 games for $20 plus $10 per child.
Estimate the total cost if 5 of Emma’s friends come to her party.
Estimate the total cost if 15 friends come to Emma’s party.
(she must be a very popular girl!)
Explain how you found the costs in #1 and 2.

11. Write an equation representing the total costs of Emma’s
birthday party at Valley Bowling Alley. Let t = the total cost of
the bowling alley party (excluding cake, decorations…) and
let n = the number of Emma’s friends who attend the party.
t =
Her mother can rent a party room, let the kids rent shoes and
play 3 games for $20 plus $10 per child.
Estimate the number of friends Emma can invite for $240.
You only need to estimate to answer this question.
Which variable do you know in #5?
Where would you put it, if you wanted to use the equation?
Try to set up the equation.

12. Which variable do you know in #1.
Estimate the total cost if 5 of Emma’s friends come to her party.
Try to set up the equation for that situation.
Estimate the total cost if 5 of Emma’s friends come to her party.
Do you think these 2 equations are the same of different? Why?

13. 3.2
Solving Linear
With
Algebraic Notation

14. When my daughter was in 3rd grade, she had a problem
that said:
What number is added to 7 to get 15.
I told her, “Emma, think 15 – 7 = ?
“15 – 7 = 8”, she said.
“Does 7 + 8 = 15?”
She thought for a minute, then “YES!”

15. What’s My Number ?
When I add 8 to my number, I get 20. What’s my number?
Can you make an equation out of that problem?
What should I tell my daughter to think?
When I divide my number by 2, I get 6. What’s my number?
Can you make an equation out of that problem?

16. When solving linear equations, we work backward
To move things across the equal sign to ultimately
get the variable by itself.
Reverse the order of operations.
What is the last order of operations?
What comes before that?

17. SOLUTION – a value for the variable that makes the equation true.
3x – 2 = 1 Check for x = 0, x = 1
2 – 6x = 2 Check for x = -1, x = 1, x = 0
SOLUTION SET– all solutions to an equation.
SOLVING – a process where you use inverse operations.

18. ADDITION PROPERTY for EQUATIONS
if a = b, then a + c = b + c.
You can ADD (SUBTRACT)the same thing to
both sides of an equation.
MULTIPLICATION PROPERTY for EQUATIONS
if a = b, then ac = bc.
You can MULTIPLY (DIVIDE)the same thing to
both sides of an equation.

19. x + 8 = 20
What do you think we need to do 1st ?
x + 8 = 20
-8 -8
x + 0 = 12
x = 12

20. White beans are positive numbers
Black beans are negative numbers
ve numbers
Candy is the variable
White and Black beans cancel each other out.
### x - ##### = -###
##### #####
### x = ##
X = ##/###

21. MAKE A PLAN
EQUATION KNOWN? ACTION CHECK
x + 3 = 2 3 is added to x
Subtract 3 from
both sides
x = -1
(-1) + 3 = 2
4
3


x
9
2
1
x
**NOTE: you can change the subtraction on the variable to addition
1. The 2 is added
to the variable term
2. x is multiplied
by –6
5 = 2 – 6x
4x = 2
Subtract 2 from
both sides
divide both
sides by -6 )( 725
6
7
625
6
7






 


x

22. Look in your book:
Page 133 #56
Page 143 #82, 83, 84

23. 3.3
Solving Equations
Tables, Graphs, Symbols

24. EQUIVALENT – when you do something to both sides it
remains equivalent
3x + 5 = 7 is equivalent to 3x = 2
(subtracted 5 both sides)
Tables and Graphs-- give us a numerical and visual
picture of what solving an equation means.

25. Solving Equations
With
Tables and Graphs
Worksheet

26. x y = 3x - 2
-2
0
1
3
Solve (by hand)
-1 = 3x - 2
0 = 3x - 2
1 = 3x - 2
3 = 3x - 2
CALCULATOR STEPS
1. Y =
2. 2nd – WINDOW
3. TBLSTART = -2
4. Δ TBL = 1/3
5. 2nd - Graph

27. x y = 4 - x
2
0
5
x y = 5 - 3x
-1
0
5
CALCULATOR STEPS
1. Y =
2. 2nd – WINDOW
3. TBLSTART = 0
4. Δ TBL = 1
5. 2nd - Graph
CALCULATOR STEPS
1. Y =
2. 2nd – WINDOW
3. TBLSTART = -1
4. Δ TBL = 1/3
5. 2nd - Graph

28. GRAPHS:
4 = 3x - 2
-5 = 3x - 2
1 = 3x - 2
y = 3x - 2
See page 147
9 = 3x - 2y=9
y=4
y=1
y=-5
CALCULATOR STEPS
1. Y1 = left side
2. Y2 = right side
3. Window (set it)
4. Graph

29. GRAPHS:
4 = 4 - x
-7 = 4 - x
1 = 4 - x
y = 4 - x
y=4
y=1
y= -7
Go to pg 153 - #10
Book/Calc.

30. (#18) 3(x– 1) = -3
Does this look like linear form yet?
What should we do 1st to get to linear form?
  34
3
1
x
(#30) 6 – 5(x – 1) = 31
**can change subtraction to adding a (-)
SYMBOLS: (page 154)
(#26)
x = 0
x = 5
x = -4

31. How does a table show the solution to the equation?
How does a graph show the solution to the equation?
Which of the 3 methods would you prefer for
5 + 9x = 21?
Which of the 3 methods would you prefer for
5(x – 7)2 + x = 20 ?
Look at y column for what is known and over to x column for answer.
Look at y axis, go across to line, then down to solution off x axis.
Table or Symbols– not a nice round number for graph
Graph or Table – a lot of work for symbols.

32. WORD PROBLEMS –page 154 # 38, 40, 42
#38 7(x + 3) = - 21
#40 1/3(x + 6) = -2
#42 70 = 10(x – 4) + 30
Mid-Chapter Review:
page 155-6 # 2abd, 3, 4, 7, 10, 15, 19ad, 20

33. 3.4
Solving Linear Equations
With
Variables on BOTH sides

34. White beans are positive numbers
Black beans are negative numbers
Candy is the variable
White and Black beans cancel each other out.
Balance Beans
Try these in your group:
3x – 1 = x + 1
5x + 6 = 3x + 2
6x – 6 = 10 – 2x
TRY: 2(x + 1) = 4x - 8
3x – 5 = -3 + x
# # # x - # # # # # = - # # # + # x
- #x___________________ - #x
## x - # # # # # = - # # #
## x - # # # # # = - # # #
+ # # # # # +# # # # #
# # x = # #
X = # #/# # = #

35. Solving
Symbolically
6 – 5x = 3(1 – x)
Always clear parentheses 1st
Collect all variables on 1 side
Add 5x to both sides-why?
Collect all constants on the other side.
Solve for x.

37. APPLICATIONS
Page 165
# 46

38. U-Haul Budget
y = 39.95 + 1.59x y = 69.99 + 0.89x
In New York City, a U-Haul 24-foot moving truck costs $39.95 plus
1.59 per mile for 6 hours. The same truck from Budget costs $69.99
plus $0.89 per mile.
State the equation by setting the two cost equations
equal to each other.
Solve and explain the meaning of the solution.
What does the output of the point of intersection mean?

39. Tables and Graphs
Worksheet
(in groups)
Look at Warm-Up
Page 166

40. 3.5
Solving Linear Inequalities
In 1 Variable

41. SYMBOLIC
FOR THE FOLLOWING, DETERMINE IF THE INEQUALITY IS
TRUE OR FALSE AFTER THE OPERATION IS DONE.
4 < 10 ADD 2 to both sides. True/False
4 < 10 ADD -2 to both sides True/False
4 < 10 MULTIPLY 2 to both sides. True/False
4 < 10 MULTIPLY -2 to both sides. True/False
4 < 10 DIVIDE 2 to both sides. True/False
Try -2. True/False

42. GENERAL RULE for INEQUALITIES
The inequality stays the same when adding and subtracting across
the inequality sign.
The inequality ONLY CHANGES when you Multiply or Divide both
sides by a negative number.

43. TABLES
x – 1 < 2
x y =x - 1 y = 2
1 0 2
2 1 2
3 2 2
4 3 2
5 4 2
Find where they are the same
Look at the behavior of the numbers before and after that point
What set of values makes the
inequality true?
Let’s graph on a number line:
0 2 4
Which side of 2 makes a true
statement? Test the points on
either side
Look at graphs

44. TABLES
5 < 2 – 3x
x y = 5 y = 2 – 3x
-3 5 11
-2 5 8
-1 5 5
0 5 2
1 5 -1
Find where they are the same
Look at the behavior of the numbers before and after that point
What set of values makes the
inequality true?
Let’s graph on a number line:
-3 -1 0
Which side of -1 makes a true
statement? Test the points on
either side
Look at graphs
Try:
3 – 3x > 2 – 2x

45. GRAPHS
Find the point of intersection
Check to see which graph has the larger y values
BEFORE the intersection
Look what happens to the y values of that line AFTER
the intersection
Which side of the intersection would make the
inequality true?

46. APPLICATIONS
The graph below represents the profit for Companies A and B if they sell x
units of their product.
Amount of
Product
Company A
Profit
Company B
Profit
20,000
45,000
Complete the following table by
using the graph to estimate.

47. 1. Which company has the
larger profit when 20,000
units are produced?
2. Which company has the
larger profit when 45,000
units are produced?
3. Approximate from the graph,
when the two companies’
profits are equal.
4. How many units must be
sold for Company A’s profits
to be larger than Company
B’s profits?

48. END OF CHAPTER 3