Copyright Scott Storla 2015 The idea of a solution.

Copyright Scott Storla 2015

The idea of a solution


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 andan equality symbol.

4 5 2 1x x 4 5 2 1x x

4 5 2 1x x

6x 4

4 5 2 1x x 2 5 1x

2 6x 3x

3

You often simplify an expression.

You often solve an equation.

You simplify using the order of operations.

You solve by building simpler equivalent equations.


The idea of a solution

Rightexpression

Left

expression

4 2x


We often think of an equation having a “left” expression and a “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.

4 5 10x

26


Originally people solved equations by guessing and checking.

Maybe it’s 1

16 10

4 4 10

4 5 10 141 2

4 5 10 4 2x x

1 is not the solution.

4 5 10 4 2Solve x x

4 2

2


Maybe it’s 4

14

16 2

4 5 10 444 2

4 is the solution.

4

4 5 10 4 2Solve x x

Originally people solved equations by guessing and checking.

4 5 10 4 2x x

4 1 10

4 10

14


Check whether the given value is a solution.

2( 5) 10 2, 2x x

2 5 1 2 22 0

4

6 10

4

Yes, 2 is the solution.

2 3 10


Check whether any supplied value is a solution.

3 1 5 4; 2 , 1 , 1x x

3 2 1 5 2 4 3 1 1 5 1 4 3 1 1 5 1 4

63(3) 5

9 5

14


5 3 0 5

0 5

5

1 is the solution.

3 3 2 5

6 5

1

-1 is not the solution.

1


Check whether the given value is a solution.

Yes, – 20 is the solution.

32; 20

9 6 36

k k k

20 23 ( )2

9 6

0 20

36

17 20 20 2

9 6 36 1

68 120 20 72

36 36 36 36

52 52

36 36


Check whether any supplied value is a solution.

( 1) 7 4 2( 1) (6) 7 4 2(6) (11) 7 4 2(11)

4 2

4 2

6

1 7

6

-1 is not the solution.

4 12

8

13


4 22

18

18

11 is the solution.

7 4 2 ; 1 , 6 , 11y y

11


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 return 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.

932

5F C

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.

932

5F C

932

5F C


An Introduction to Formulas


Choosing the right formula


Temperature

932

5F C 5 32

9

FC

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

932

5F C

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

5 32

9

FC

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

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 350º Fahrenheit?


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.

Paying Off a Loan

6,500 250A t 26 0.004t A

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


Temperature

932

5F C 5 32

9

FC

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

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

5 32

9

FC

5 32 32

9C

5 0

9C

0C

Water freezes at 0° Celsius.

0

9C

932

5F C 5 32

9

FC


Temperature

932

5F C

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

932

5F C

960 32

5F

108 32F

140F

At 140° Fahrenheit it would be too hot.

5 32

9

FC


Temperature

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?

920 32

5F

9 4 32F

36 32F

You should set the temperature at 68° Fahrenheit.

68F

932

5F C

932

5F C 5 32

9

FC


Temperature

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 350º Fahrenheit?

5 32

9

FC

5 350 32

9C

5 318

9C

177C

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

1590

9C

932

5F C 5 32

9

FC


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.

Circumference and Diameter

C D CD


A lake that is approximately circular has a diameter of 2.8 miles. Approximately how many miles of shore line does it have?

3.14(2.8)

8.8

C D

C

C

The lake has approximately 8.8 miles of shore line.


C D CD

3.14(2.8)

8.8

C D

C

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.

3.14(7918)

24,863

C D

C

C

You would need about 24,863 miles of rope.


C D CD

3.14(7918)

24,863

C D

C

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?

43.14

1.3

CD

D

D

She swims about 1.3 miles.


C D CD

43.14

1.3

CD

D

D


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?

603.14

19.1

CD

D

D

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


C D CD

603.14

19.1

CD

D

D


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

6,500 250

6,500 250(12)

6,500 3,000

3,500

A t

A

A

A

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

Paying Off a Loan

6,500 250A t 26 0.004t A

6,500 250

6,500 250(12)

6,500 3,000

3,500

A t

A

A

A


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

26 0.004

26 0.004(1,000)

26 4

22

t A

t

t

t

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

Paying Off a Loan

6,500 250A t 26 0.004t A

26 0.004

26 0.004(1,000)

26 4

22

t A

t

t

t


How long will it take to pay off the loan?

26 0.004

26 0.004(0)

26 0

26

t A

t

t

t

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

Paying Off a Loan

6,500 250A t 26 0.004t A

26 0.004

26 0.004(0)

26 0

26

t A

t

t

t


What was the original amount of the loan?

6,500 250

6,500 250(0)

6,500 0

6,500

A t

A

A

A

The original loan amount was $6,500.

Paying Off a Loan

6,500 250A t 26 0.004t A

6,500 250

6,500 250(0)

6,500 0

6,500

A t

A

A

A


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.

Life Expectancy

0.2 70L t 5 70t L


Estimate the life expectancy of someone born in 2020.

0.2 70

0.2(60) 70

12 70

82

L t

L

L

L

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

Life Expectancy

0.2 70L t 5 70t L

0.2 70

0.2(60) 70

12 70

82

L t

L

L

L


Estimate the life expectancy of someone born in 1960.

0.2 70

0.2(0) 70

0 70

70

L t

L

L

L

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

Life Expectancy

0.2 70L t 5 70t L

0.2 70

0.2(0) 70

0 70

70

L t

L

L

L


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

5 70

5 85 70

5 15

75

1960 75 2035

t L

t

t

t

In 2035 life expectancy will first reach 85 years.

Life Expectancy

0.2 70L t 5 70t L

5 70

5 85 70

5 15

75

1960 75 2035

t L

t

t

t


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

0.2 70

0.2(25) 70

5 70

75

L t

L

L

L

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

Life Expectancy

0.2 70L t 5 70t L

5 70

5 76.5 70

5 6.5

32.5

1960 32 1992

t L

t

t

t

0.2 70

0.2(25) 70

5 70

75

L t

L

L

L

5 70

5 76.5 70

5 6.5

32.5

1960 32 1992

t L

t

t

t


Applying Formulas


An introduction to formulas

Copyright Scott Storla 2015 The idea of a solution.

Documents

solutioncopyright scott

celsius temperature

fahrenheit temperature

temperature fahrenheit

right expression

left expression

supplied value

given value