1. An Introduction to Equations
Section 1-8

2. Goals Goal Rubric To solve equations using tables and mental math.
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.

3. Vocabulary
Equation
Open sentence
Solution of an equation

4. 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

5. 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

6. 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.

7. Example: Classifying Equations
Is the equation true, false, or open? Explain.
3y + 6 = 5y – 8
Open, because there is a variable.
16 – 7 = 4 + 5
True, because both sides equal 9.
32 ÷ 8 = 2 ∙ 3
False, because both sides are not equal, 4 ≠ 6.

8. Your Turn: Is the equation true, false, or open? Explain.=
False, because both sides are not equal, 26 ≠ 25.
4 ∙ 11 = 44
True, because both sides equal 44.
3x – 1 = 17
Open, because there is a variable.

9. 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.

10. Example: Identifying Solutions of an Equation
Is m = 2 a solution of the equation 6m – 16 = -4? 6m – 16 = -4 6(2) – 16 = – 16 = = -4 True statement, m = 2 is a solution.

11. Your Turn:
Is x = 5 a solution of the equation 15 = 4x – 4? No, 15 ≠ 16. False statement, x = 5 is not a solution.

12. 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.
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.
Assign labels to each part of your verbal problem.
ALGEBRAIC
MODEL
Use the labels to write an algebraic model based on your verbal model.
Use the labels to write an algebraic model based on your verbal model.

13. 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.

14. – – • = = = The equation is 2p = 24. Cost per plate = 2 p
Writing an Equation
VERBAL
MODEL
Cost per
plate
Number of plates • = –
Bill
Tax
LABELS
Cost per plate = 2
(dollars)
ALGEBRAIC
MODEL p
Number of plates =
(plates)
Amount of bill =
25.20
(dollars)
Tax =
1.20
(dollars) 2 = p
25.20
1.20 – 2p =
24.00
The equation is 2p = 24.

15. 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?

16. 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
Distance to
travel •
Time =
LABELS
Speed of jet = x
(miles per hour)
ALGEBRAIC
MODEL
Time = 2
(hours)
Distance to travel =
600
(miles) 2 x =
600 2x =
600

17. 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.

18. Your Turn: What is the solution to the equation? Use mental math.
x + 7 = 13 6
x/6 = 12 72

19. 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.

20. Assignment
1.8 Exercises Pg. 64 – 66: #8 – 32 even, 48 – 52 even, 56 – 74 even