Building Algebraic Expressions

The idea of a solution

Copyright Scott Storla 2015

The difference between an expression and an equation


Expression vs. Equation

An expression is a meaningful collection of numbers, letters, operators and grouping symbols.

An equation is a meaningful collection of numbers, letters, operators grouping symbols and an equality symbol.

4 x 5 2 x



1

4 x



5



2 x



1

You often simplify an expression.

You often solve an equation.

You simplify using the order of operations.

4 x 5 2 x



1

6 x



4

You solve by building simpler equivalent equations.

4 x



5 2 x



1

2 x



5





1

2 x x





6

3


A woman is preparing for a triathlon.

She runs 7 miles every day. If her morning run is 4 miles further than her evening run, how far does she run in the evening?

x



7 ( 4)


The idea of a solution


We often think of an equation having a

“left” expression and a “right” expression.

4

 x



5

 

10

Left

expression

 

4 x



2

Right expression

To solve an equation you find all values that, after substituting and simplifying, makes the value of the left expression and the value of the right expression the same.


Solve 4

 x 5



10

 

4 x



2

Originally people solved equations by guessing and checking.

4

 x 5



10

 

4 x



2

4



1



10



4



2

4

 

10 4 2



2



26

1 is not the solution.


Maybe it’s

1

Solve 4

 x



5

 

10

 

4 x



2

Originally people solved equations by guessing and checking.

4

 x



5

 

10

 

4 x



2

4



4



5

 

10



4



2

4

 

10 16 2



14



14

4 is the solution.


Maybe it’s

4

Check whether the given value is a solution.

2( x



5) 10 x

2



2



5

 

1 0

2

 

10

4

2,

2



2

4

Yes, 2 is the solution.


Check whether any supplied value is a solution.

 

5 2



4

6

3



5

8

3

 x 1



5


4;

     

 

5



 



5 5

0



5

5

1 is the solution.

3



1 1



5

3

 

5

6 5



1

-1 is not the solution.

3


Check whether the given value is a solution.

3

 k k k

9 6 36



2;



9



20

2 6

18

)







17





20

9 6



34





60

18 18

20

6



2 0



2

36



20



2

36 1



20



72

36 36

52

36



2

2

 



26

18

Yes, – 20 is the solution.


Check whether any supplied value is a solution.

   

2 ;

     

( 1) 7

 



6

4

 

2

4



2

6

-1 is not the solution.



(6) 7 4



2(6)



13



8



(11) 7





18



18


11 is the solution.


The Idea of a Solution


An introduction to formulas


For now, we’ll think of a function as a process that uses a value we supply to find a value we’re interested in.

A formula is a function that uses algebra to express a fact or rule.

This is an example of a formula.

F



9

5

C



32

This formula gives us a way to operate on a

Celsius temperature and find the Fahrenheit temperature.


We’ll write a formula using the equality symbol, = , to separate two expressions into a “left side” and a

“right side”.

On the left side we write a single variable that holds the place of the value we’re looking for.

On the right side we write a mathematical expression that might include numbers, variables, operators and the idea of grouping.

F



9

5

C




An Introduction to Formulas


Choosing the right formula


F



9

5

C



32

Temperature

C



5



F



32



9

Fahrenheit is the temperature scale named after physicist Daniel Fahrenheit. It’s the official temperature scale for the United States.

Celsius (centigrade) is the temperature scale named after astronomer Anders Celsius. It’s the official scale for most of the rest of the world.

We’ll use F to represent a temperature in degrees

Fahrenheit and C to represent a temperature in degrees Celsius.


Temperature

F



9

5

C



32 C



5



F



32



9

Water freezes at 32 º Fahrenheit. At what Celsius temperature does water freeze?

Would a hot tub that’s 60º Celsius be too hot or too cold?

A passenger from Canada says they prefer the temperature in your car to be around 20 degrees.

To accommodate their needs what Fahrenheit temperature should you use?

On a trip to Europe you plan to cook a dish for your relatives. What Celsius temperature should you use if you need to set the oven to 350 º Fahrenheit?


Paying Off a Loan

A



6,500



250 t t



26 0.004

A

If A stands for the amount still owed on a loan (in dollars) and t stands for the number of months payments have been made on the loan. Decide which formula you should use to answer the following questions.

How much will you still owe on the loan after paying for 1 year?

How long will it take to get the loan down to around $1,000?

How long will it take to pay off the loan?

What was the original amount of the loan?


Choosing the Right Formula


Applying Formulas


F



9

5

C



32

Temperature

C



5



F



32



9

Fahrenheit is the temperature scale named after physicist Daniel Fahrenheit. It’s the official temperature scale for the United States.

Celsius (centigrade) is the temperature scale named after astronomer Anders Celsius. It’s the official scale for most of the rest of the world.

We’ll use F to represent a temperature in degrees

Fahrenheit and C to represent a temperature in degrees Celsius.


Temperature

F



9

5

C



32 C



5



F



32



9

Water freezes at 32 º Fahrenheit. At what Celsius temperature does water freeze?

C

C





5



F



32



9

 

9

C



0

Water freezes at 0 ° Celsius.

C



9

C



0

9


F



9

5

C



32

Temperature

C



5



F



32



9

Would a hot tub that’s 60º Celsius be too hot or too cold?

F



9

5

C



32

F



9

5

  

32

F



108



32

F



140

At 140 ° Fahrenheit it would be too hot.


Temperature

F



9

5

C



32 C



5



F



32



9

A passenger from Canada says they prefer the temperature in your car to be around 20 degrees. To accommodate their needs what Fahrenheit temperature should you use?

F



9

5

C



32

F



9

5

  

32

F



 



32

F



36



32

F



68

You should set the temperature at 68 ° Fahrenheit.


Temperature

F



9

5

C



32 C



5



F



32



9

On a trip to Europe you plan to cook a dish for your relatives. What Celsius temperature should you use if you need to set the oven to 350 º Fahrenheit?

C



5



F



32



9

C





C



9

 

9



C



177

You should set the temperature of the oven to 177 ° C .

C



1590

9


Circumference and Diameter

C

 

D D

 C



Before algebra it was often difficult to measure the distance of the boundary of a circle (the circumference ).

Many cultures tried to find a way to measure the circumference using a straight line. One intuitive straight line distance for circles is to measure from one edge to the other while going through the center of the circle (the diameter ).

Use C for the distance of the circumference, D for the distance of the diameter and 3.14 for the value of pi.

If necessary, round to the tenths place so our answers will be consistent.



C

 

D D

 C



A lake that is approximately circular has a diameter of

2.8 miles. Approximately how many miles of shore line does it have?

C

 

D

The lake has approximately 8.8 miles of shore line.



C

 

D D

 C



The diameter of the Earth is approximately 7,918 miles.

How many miles of rope would you need to circle the

Earth if the rope was lying on the ground? Round to the nearest mile.

C

 

D

You would need about 24,863 miles of rope.



C

 

D D

 C



To train for a triathlon a woman runs around a lake with a circumference of approximately 4 miles and then swims across the lake from shore to shore. If her swim takes her through the middle of the lake, about how far does she swim?

She swims about 1.3 miles.



C

 

D D

 C



A pizza restaurant decides to create a “super-sized” pizza with a circumference just under 60 inches. If pizza boxes only come with side lengths that are natural numbers what side length would work for their super-sized pizza?



19.1

The box needs to have a side length of 19 inches.


Paying Off a Loan

A



6,500



250 t t



26 0.004

A

How much will you still owe on the loan after paying for 1 year?

6,500



250 t t

6,500



250(12)

$3,500 is still owed after paying for 1 year.


Paying Off a Loan

A



6,500



250 t t



26 0.004

A

How long will it take to get the loan down to around $1,000? t t t t



26 0.004

t t



26 0.004(1,000) t t



26



4

It will take about 22 months for the loan to drop to $1,000.


Paying Off a Loan

A



6,500



250 t t



26 0.004

A

How long will it take to pay off the loan?

t t t t



26 0.004

t t



26 0.004(0) t t



26 0

It will take 26 months for the loan to drop to $0.


Paying Off a Loan

A



6,500



250 t t



26 0.004

A

What was the original amount of the loan?

6,500



250 t

6,500



250(0)

The original loan amount was $6,500.


L



0.2

t



70

Life Expectancy t



5



L



70



L is an estimate of how long the average U.S. citizen will live if you supply t the number of years after 1960 they were born. For example for someone born in 1994 t would have the value 34 (34 years after 1960). Use the proper formula to answer the following questions.


L



0.2

t



70

Life Expectancy t



5



L



70



Estimate the life expectancy of someone born in 2020.

0.2

t t



70



12 70

Someone born in 2020 will have a life expectancy of 82 years.


L



0.2

t



70

Life Expectancy t



5



L



70



Estimate the life expectancy of someone born in 1960.

0.2

t



70

0 70

Someone born in 1960 had a life expectancy of 70 years.


L



0.2

t



70

Life Expectancy t



5



L



70



Estimate the first year that someone will have an average life expectancy of 85 years.

t t t t 5

 

L



70

 

    t t t t

1960



75



2035

In 2035 life expectancy will first reach 85 years.


L



0.2

t



70

Life Expectancy t



5



L



70



Estimate the first year that someone will have a life expectancy 1.5 years above the 1980 level.

0.2

t t



70 t t t t

5

 

L



70

 

   

5 70 t t t t

1960



32



1992

The first year that will occur is around 1992-1993.


Applying Formulas


An introduction to formulas


Building Algebraic Expressions

Related documents

Products

Support

Building Algebraic Expressions

Related documents

Add this document to collection(s)

Add this document to saved

Suggest us how to improve StudyLib