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An Introduction to Equations

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An Introduction to Equations

Section 1-8

Goal

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.

- To solve equations using tables and mental math.

- Equation
- Open sentence
- Solution of an equation

- 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

Equations or Inequalities

Sentences

- 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.

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.

Is the equation true, false, or open? Explain.

- 17 + 9 = 19 + 6
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.

- 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.

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.

Is x = 5 a solution of the equation 15 = 4x – 4?

No, 15 ≠ 16. False statement, x = 5 is not a solution.

LABELS

ALGEBRAIC

MODEL

Procedure for Writing an Equation

A PROBLEM SOLVING PLAN USING MODELS

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.

VERBAL

MODEL

Assign labels to each part of your verbal problem.

Assign labels to each part of your verbal problem.

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.

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

Cost per

plate

Number of plates

Bill

LABELS

p

Tax

25.20

2

2

ALGEBRAIC

MODEL

–

p

=

1.20

25.20

1.20

The equation is 2p = 24.

VERBAL

MODEL

–

=

•

Cost per plate =

(dollars)

Number of plates =

(plates)

Amount of bill =

(dollars)

Tax =

(dollars)

=

2p

24.00

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?

Speed of

jet

Time

You can use the formula (rate)(time) = (distance) to write a verbal model.

LABELS

2

x

600

ALGEBRAIC

MODEL

2

x

600

Solution

At what speed would the jet have to fly to arrive in Chicago in 2 hours?

SOLUTION

Distance to

travel

VERBAL

MODEL

=

•

Speed of jet =

(miles per hour)

Time =

(hours)

Distance to travel =

(miles)

=

2x

=

600

- 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.

What is the solution to the equation? Use mental math.

- x + 7 = 13
6

- x/6 = 12
72

- 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.

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