An Introduction to Equations
Section 1-8
Goals
Goal
• To solve equations using
tables and mental math.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• Equation
• Open sentence
• Solution of an equation
Definition
• Equation – A mathematical sentence that states one
expression is equal to a second expression.
•
mathematical sentence that uses an equal sign (=).
• (value of left side) = (value of right side)
• An equation is true if the expressions on either side of the
equal sign are equal.
• An equation is false if the expressions on either side of the
equal sign are not equal.
• Examples:
• 4x + 3 = 10 is an equation, while 4x + 3 is an expression.
• 5 + 4 = 9 True Statement
• 5 + 3 = 9 False Statement
Equation or Expression
In Mathematics there is a difference between a phrase
and a sentence. Phrases translate into expressions;
sentences translate into equations or inequalities.
Phrases
Expressions
Sentences
Equations or Inequalities
Definition
• Open Sentence – an equation that contains
one or more variables.
– An open sentence is neither true nor false until
the variable is filled in with a value.
• Examples:
– Open sentence: 3x + 4 = 19.
– Not an open sentence: 3(5) + 4 = 19.
Example: Classifying
Equations
Is the equation true, false, or open? Explain.
A. 3y + 6 = 5y – 8
Open, because there is a variable.
B. 16 – 7 = 4 + 5
True, because both sides equal 9.
C. 32 ÷ 8 = 2 ∙ 3
False, because both sides are not equal, 4 ≠ 6.
Your Turn:
Is the equation true, false, or open? Explain.
A. 17 + 9 = 19 + 6
False, because both sides are not equal, 26 ≠ 25.
B. 4 ∙ 11 = 44
True, because both sides equal 44.
C. 3x – 1 = 17
Open, because there is a variable.
Definition
• Solution of an Equation – is a value of the
variable that makes the equation true.
– A solution set is the set of all solutions.
– Finding the solutions of an equation is called
solving the equation.
• Examples:
– x = 5 is a solution of the equation 3x + 4 = 19,
because 3(5) + 4 = 19 is a true statement.
Example: Identifying
Solutions of an Equation
Is m = 2 a solution of the equation
6m – 16 = -4?
6m – 16 = -4
6(2) – 16 = -4
12 – 16 = -4
-4 = -4 True statement, m = 2 is a solution.
Your Turn:
Is x = 5 a solution of the equation
15 = 4x – 4?
No, 15 ≠ 16. False statement, x = 5 is not a
solution.
Procedure for Writing an Equation
A PROBLEM SOLVING PLAN USING MODELS
VERBAL
MODEL
Ask yourself what you need to know to solve the
problem. Then write a verbal model that will give
you what you need to know.
LABELS
Assign labels to each part of your verbal problem.
ALGEBRAIC Use the labels to write an algebraic model based on
MODEL
your verbal model.
Writing an Equation
You and three friends are having a dim sum lunch at a Chinese
restaurant that charges $2 per plate. You order lots of plates.
The waiter gives you a bill for $25.20, which includes tax of
$1.20. Write an equation for how many plates your group
ordered.
SOLUTION
Understand the problem situation
before you begin. For example,
notice that tax is added after the total
cost of the dim sum plates is figured.
Writing an Equation
VERBAL
MODEL
Cost per
plate
LABELS
Cost per plate = 2
(dollars)
Number of plates = p
(plates)
•
Number of
plates
=
Bill
Amount of bill = 25.20 (dollars)
Tax = 1.20
ALGEBRAIC
MODEL
2
p = 25.20
2p = 24.00
The equation is 2p = 24.
(dollars)
–
1.20
–
Tax
Your Turn:
JET PILOT A jet pilot is flying from Los Angeles, CA to Chicago, IL at a
speed of 500 miles per hour. When the plane is 600 miles from Chicago,
an air traffic controller tells the pilot that it will be 2 hours before the
plane can get clearance to land. The pilot knows the speed of the jet must
be greater then 322 miles per hour or the jet could stall.
Write an equation to find at what
speed would the jet have to fly to
arrive in Chicago in 2 hours?
Solution
At what speed would the jet have to fly to arrive in Chicago in 2 hours?
SOLUTION
You can use the formula (rate)(time) = (distance) to write a verbal model.
VERBAL
MODEL
Speed of
jet
LABELS
Speed of jet = x
(miles per hour)
Time = 2
(hours)
Distance to travel = 600
(miles)
ALGEBRAIC
MODEL
•
Time
2 x = 600
2x = 600
=
Distance to
travel
Example: Use Mental Math to
Find Solutions
• What is the solution to the equation? Use
mental math.
• 12 – y = 3
– Think: What number subtracted from 12 equals 3.
– Solution: 9.
– Check: 12 – (9) = 3, 3 = 3 is a true statement,
therefore 9 is a solution.
Your Turn:
What is the solution to the equation? Use mental
math.
A. x + 7 = 13
6
B. x/6 = 12
72
Joke Time
• What do you call a guy with a rubber toe?
• ROBERTO.
• What do you get if you cross a dinosaur with a
pig?
• Jurassic Pork.
• What do you get when you put a bomb and a
dinosaur together?
• Dino-mite.
Assignment
• 1.8 Exercises Pg. 64 – 66: #8 – 32 even, 48
– 52 even, 56 – 74 even