elementary algebra exam 2 material. equations equation – a statement that two expressions are...

Elementary Algebra

Exam 2 Material

Equations

• Equation – a statement that two expressions are equal– Equations always contain an equal sign, but an

expression does not have an equal sign

• Like a statement in English, an equation may be true or false

• Examples:

.

1495 F?or T

574

True

F?or T False

Equations

• Most equations contain one or more variables and the truthfulness of the equation depends on the numbers that replace the variables

• Example:

• What value of x makes this true?

• A number that can replace a variable to make an equation true is called a solution

94 x5x

equation theosolution t a is 5

Types of Equations

• In Algebra you will study many different types of equations– Learn the names of each type– Learn method for solving each type

• The simplest type of equation is called a “linear equation”

Linear Equations

• Linear equation – an equation where, after parentheses are gone, every term is either a constant, or of the form: cx where c is a constant and x is a variable with exponent1Linear equations never have a variable in a denominator or under a radical (square root sign)

• Examples of Linear Equations:

.

173 xx

xx5

332 1354 x

xxx 382

1627.

Identifying Linear Equations

• Identify linear equations:

82

1

3

7

x

x

22 xx

3

7

2

5

x

x

914 xx

No

No

No

Yes

e)on variabl 2(exponent

r)denominatoin (variable

radical)under (variable

cx)or c is (each term

Solving Linear Equations

• Any equation that is a true statement has both sides with equal values. It is “balanced on both sides of the equal sign.”

• In trying to find solutions to an equation we try to do things that will keep both sides of an equation balanced while progressing towards a goal of ending up with the variable alone on one side of the equal sign

Solving Linear Equations by Keeping Both Sides Balanced

• If we assume that both sides of an equation really are equal and we add or subtract the same thing on both sides, then both sides will still be balanced

• In the earlier example, ,what could we have done on both sides of original equation to get a new equation with “x” isolated?

94 x

4)Subtract (or 4- Add

4944 x5x

Solving an Equation by Balancing with Addition or Subtraction

• Solve the equation:

• The solution to the original equation is:

129 x

91299 x

3x

3x

Solving More Complicated Equations

• In solving an equation, make sure that the expression on each side is simplified, before proceeding – Get rid of parentheses– Combine like terms

• Next, choose which side will keep the variable and add or subtract terms on both sides, as necessary, to isolate the variable

.

24532 xxx84562 xxx

862 xx862 xxxx

86 x6866 x

2x

English Sentences that Translate to Equations

• If an English sentence indicates that two numerical expressions are equal, it can be translated to an equation

• The phrases: “the result is”, or “is equal to”, “equals” or “is”, can usually be translated into algebra as an equal sign, =

• Translate: The sum of a number and three is 15. 153 x

Translating Sentences to Equations and Solving

• The major reason for translating a sentence to an equation is to help us find the value of an unknown number described in the sentence

• Find the unknown number in the previous example: “The sum of a number and 3 is 15”: 153 x

31533 x12x

12. bemust describednumber The

Other Examples of Translating English Sentences to Equations

• Translate and Solve: The product of 3 and a number is equal to twice the number plus 7.

723 xx72223 xxxx

7x7. bemust describednumber The


• Translate and Solve: Twice the sum of a number and 5 is the same as 8 less than the number. What is the number?

852 xx

8102 xxxx810 x

18.- bemust describednumber The

8102 xx

1081010 x18x

Homework Problems

• Section: 2.1

• Page: 100

• Problems: Odd: 5 – 43, 47 – 65,

All: 69 – 72

• MyMathLab Section 2.1 for practice

• MyMathLab Homework Quiz 2.1 is due for a grade on the date of our next class meeting

Another Way to Keep Equations Balanced

• We have learned that equations that are true statements can be kept balanced by adding or subtracting the same thing on both sides of equal sign

• We can also keep both sides balanced by multiplying or dividing both sides by any number that is not zero

• Examples follow that show how multiplying or dividing can help solve an equation

Solving Equations Using Multiplication or Division

• Solve:

• The equation will be solved when we have

• What could we multiply on both sides?

53

1x

xnotitselfbyx3

1,

3

533

13

x

15x

Solving Equations Using Multiplication or Division

• Solve:

• The equation will be solved when we have

• What could we divide on both sides?

122 x

xnotitselfbyx 2, 2

6x

2

12

2

2

x


• Translate: The difference between 4 and a number equals the number plus 5.

.2

1- bemust describednumber The

54 xx54 xxxx

524 xx21

2

2

2

1 x

sidesboth from subtracted 5


• Translate: If 11 times a number is subtracted from 8 times the number, the result is -9.

3. bemust describednumber The

9118 xx93 x

3

9

3

3

x

3x

Homework Problems

• Section: 2.2

• Page: 107

• Problems: Odd: 7 – 69,

All: 73 – 76



Solving Linear Equations

• Simplify each side separately– Get rid of parentheses– Multiply by LCD to get rid of fractions and decimals – Combine like terms

• Get the variable by itself on one side by adding or subtracting the same terms on both sides

• If the coefficient of the variable term is not 1, then divide both sides by the coefficient

Determine if the equation is linear. If it is, solve it:

84132 xx

8426 xx

844246 xxxx

822 x28222 x

62 x

2

6

2

2 x

3x

linear?it Is Yes


427568 xx8273068 xx

12386 xx1223826 xxxx

1388 x38138388 x

378 x

8

37

8

8x

8

37x

linear?it Is Yes


1413342 xx

143368 xx17368 xx

1798 x

x99 x 1

linear?it Is Yes


• We can’t solve this equation yet. Later we will learn its name, and how to solve it!

4532 2 xxx

linear?it Is No


13.024

1

3

2 xxx

7

90x

110

3

2

1

4

1

3

2 xxx

1

10

360

2

1

4

1

3

260 xxx

6018301540 xxx

60183025 xx

60307 x

907 x

linear?it Is Yes

Example

• Solve:

• LCD:

• Multiply both sides by LCD:

24

3

6

5 xx

64LCD

3222

322 12

24

312

6

512 xx

24910 xx

24x

Example

• Solve:

• LCD:


19

53

24

7 xx

249LCD

3222 33

72

724021621 xxx

19

288

33222

1

9

5723

24

772 xx

7219216 x

x19288

Example

• Solve:

• LCD:


13.52.73. xx

10100LCD

525522

100

1322700 xx

22

713

5522 multiplelargest thealways is

10 of multiples of LCD The

13.52.10073.100 xx135270030 xx

x22713

Example with Fractions & Decimals

• Solve:

• LCD:


53.4

3

6

1 xx

6

10LCD

32

5260

300184510 xxx

8

345

5322

300845 x

x8345

4 22

53.604

3

6

160

xx

Linear Equations with No Solution or All Real Numbers as Solutions

• Many linear equations only have one number as a solution, but some have no solution and others have all numbers as solutions

• In trying to solve a linear equation, if the variable disappears (same variable & coefficient on both sides) and the constants that are left make a statement that is:– false, the equation has “no solution” (no number can

replace the variable to make a true statement)– true, the equation has “all real numbers” as solutions

(every real number can replace the variable to make a true statement)

Solve the Linear Equation

732 xxx

732 xxx

73 xx

73 xxxx

73 False!

solution no hasEquation

Solve the Linear Equation

xxxx 312272

True!

solutions are numbers real All

xxxx 62272

2828 xx

288288 xxxx

22

Homework Problems

• Section: 2.3• Page: 115• Problems: Odd: 7 – 45• Page: 117• Problems: Odd: 1 – 29

• MyMathLab Section 2.3 for practice• MyMathLab Homework Quiz 2.3 is due for a

grade on the date of our next class meeting

Solving Application Problems

• Some problems may involve more than one sentence and more than one unknown

• Such problems may seem as overwhelming as trying to eat an elephant

• How do you eat an elephant?One bite at a time!

• Application problems are solved easily if you memorize the steps and do them one at a time!

Steps in SolvingApplication Problems

• Read the problem carefully trying to understand what the unknowns are (take notes, draw pictures, don’t try to write equation until all other steps below are done )

• Make word list that describes each unknown• Assign a variable name to the unknown you know the

least about (the most basic unknown)• Write expressions containing the variable for all the other

unknowns• Read the problem one last time to see what information

hasn’t been used, and write an equation about that• Solve the equation (make sure that your answer makes

sense, and specifically state the answer)

Example of Solving anApplication Problem

• Three less than 5 times a number is equal to 9 less than twice the number. What is the number?

• List of unknowns– A number

• What else does the problem tell us that we haven’t used?Three less than 5 times a number is equal to 9 less than twice the number.

• What equation says this?

:it call wesounknown only theis This

x

9235 xx

Example Continued


• Answer to question?

The number described is:

2x

2

9235 xx922325 xxxx

933 x39333 x

63 x

Example of Solving an Application Problem With Multiple Unknowns

• A mother’s age is 4 years more than twice her daughter’s age. The sum of their ages is 76. What is the mother’s age?

• List of unknowns– Mother’s age– Daughter’s age

• What else does the problem tell us that we haven’t used?Sum of their ages is 76


about?least know wedoWhich age sDaughter'x

42 x

7642 xx

Example Continued



Mother’s age is 2x + 4:

7642 xx7643 x

476443 x723 x24x

4242 52

Solve the Application Problem

• A 31 inch pipe needs to be cut into three pieces in such a way that the second piece is 5 inches longer than the first piece and the third piece is twice as long as the second piece. How long should the third piece be?

1. Read the problem carefully taking notes, drawing pictures, thinking about formulas that apply, making charts, etc.Perhaps draw a picture of a pipe that is labeled as 31 inches with two cut marks dividing it into 3 pieces labeled first, second and third

1st 2nd 3rd

31

Example Continued

2. Read problem again to make a “word list” of everything that is unknown

What things are unknown in this problem?

The length of all three pieces (even though the problem only asked for the length of the third).

Word List of Unknowns:

Length of first

Length of second

Length of third

Example Continued

3. Give a variable name, such as “x” to the “most basic unknown” in the list (the thing that you know least about)

What is the most basic unknown in this list?

Length of first piece is most basic, because problem describes second in terms of the first, and third in terms of second

Give the name “x” to the length of first

Example Continued

4. Give all other unknowns in the word list an algebraic expression name that includes the variable, “x”How would the length of the second be named?x + 5How would the length of the third be named?2(x + 5)Word List of Unknowns: Algebra Names:Length of first xLength of second x + 5Length of third 2(x + 5)

Example Continued

5. Read the problem one last time to determine what information has been given, or implied by the problem, that has not been used in giving an algebra name to the unknowns and use this information to write an equation about the unknownsWhat other information is given in the problem that has not been used?Total length of pipe is 31 inchesHow do we say, by using the algebra names, that the total length of the three pieces is 31?x + (x + 5) + 2(x + 5) = 31

Example Continued

6. Solve the equation and answer the original questionThis is a linear equation so solve using the appropriate steps:x + (x + 5) + 2(x + 5) = 31 x + x + 5 + 2x + 10 = 31 4x + 15 = 31 4x = 16 x = 4Is this the answer to the original question?No, this is the length of the first piece.How do we find the length of the third piece?The length of the third piece is 2(x + 5):2(4 + 5) = (2)(9) = 18 inches = length of third piece

Solving Application Problems Involving Angles

• Angles are measured in units called degrees• A complete rotation of a ray from a starting position back to the

starting position has a measure of 360o

• Half of a rotation of a ray from a starting position to a position pointing the opposite direction has a measure of 180o and the angle is called a straight angle

• One quarter of a rotation of a ray has a measure of 90o and the angle is called a right angle

• Two angles whose sum makes a right angle (whose sum is 90o) are called complementary angles

• Two angles whose sum makes a straight angle (whose sum is 180o) are called supplementary angles

• If x represents the measure of an angle,– The measure of the complementary angle is:– The measure of the supplementary angle is:

x90 x180

Example of Solving an Angle Application Problem

• Find the measure of an angle such that the supplement is 15o more than twice the complement

• List of unknowns– Angle measure– Complement measure– Supplement measure

• What else does the problem tell us that we haven’t used?Supplement is 15o more than twice the complement


about?least know wedoWhich angle of Measure

15902180 xx

x90x

x180

Example Continued



The measure of the angle is:

152180180 xx152 xx

15x

o15

15902180 xx

Solving Application Problems Involving Consecutive Integers

• If an application problem involves consecutive integers, consecutive even integers, or consecutive odd integers, remember that consecutive integers differ by 1 and consecutive even, as well as consecutive odd, integers differ by 2

• If x represents an integer, the next integer is x + 1, and the next is x + 2, etc

• If x represents an even integer, the next even integer is x + 2, and the next is x + 4, etc

• If x represents an odd integer, the next odd integer is x + 2, and the next is x + 4, etc

Example of Solving a Consecutive Integer Application Problem

• Find two consecutive odd integers such that three times the smaller is thirteen less than twice the larger

• List of unknowns– smaller odd integer– next larger odd integer

• What else does the problem tell us that we haven’t used?Three times smaller is 13 less than twice larger


about?least know wedoWhich integer oddFirst

x2x

13223 xx

Example Continued


• Answer to question?The smaller odd integer is x =The larger odd integer is x + 2 =

13423 xx923 xx

9x

9

13223 xx

7

Homework Problems

• Section: 2.4

• Page: 126

• Problems: Odd: 11 – 57



elementary algebra exam 2 material. equations equation – a statement that two expressions are...

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