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ALGEBRA A Self-Tutorial by

Luis Anthony Ast Professional Mathematics Tutor

INTRODUCTION TO ALGEBRA Copyright Š 2006 All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing of the author.

E-mail may be sent to: Luis-Ast@VideoMathTutor.com


BEFORE YOU GET STARTED‌ It is highly recommended you first take the time to review the BASIC MATH lessons that are available on the web site. To do well in algebra, you should have a good understanding of arithmetic.

OK, NOW LET ’S GET STARTED F Algebra is the area of mathematics that generalizes the concepts and rules of arithmetic using symbols to represent numbers. The symbols used are called variables and constants.

F

The symbols used are usually letters from the

English alphabet. Letters from other alphabets (notably the Greek!) are also used to represent numbers.

EXAMPLE 1: The following are some examples of symbols used in algebra:

x,

,

A,

,

e,

z

F A Variable is a symbol that can represent a value that is not known or a symbol that can take on different values from a given set of values. Traditionally, the symbols most often used to represent variables are the lower case letters at the end of the alphabet: s, t, u, v, w, x, y, and z. In print, (like in these notes), variables will always be italicized. The letter x is, by far, the most commonly used variable. Contrary to what many students believe, it is not evil. It is just a variable.

=

Any letter may be used to represent a variable. It all depends on how it is being used in a problem.

F A Constant is a number or symbol that represents a specific value. 2


The value of π (pi) is the most well known mathematical constant. It represents the number of diameters that can fit on the circumference of a circle. Letters can be used to represent constant values. Traditionally, they are selected from the beginning of the alphabet. They can be either lower or upper case letters: a, b, c, d, e, A, B, C, D, E. In print, (like in these notes), constants will always be italicized.

SOME CONVENTIONAL NOTATION Y A constant being added to a variable is usually written to the right of the variable. <Although, technically, it really does not make a difference>

EXAMPLE 2: Here are some examples of variables and constants being added together: : x+3

w+π

y+b

The x, y, and w are the variables. The 3, b, and π are the constants. Y A constant being multiplied by a variable is usually written to the left of the variable. <Again, technically, it really does not make a difference>

EXAMPLE 3: Here are some examples of constants multiplied by variables: 5w The 2, 5, and A are the constants. The z, w, and x are the variables.

L ✓ OK: ✗ AVOID:

When multiplying variables and constants, do not use the “×” symbol. It can be confused with the letter “x.” Use parentheses, a raised dot, or just place the constants next to the variable for implied multiplication. 3x

3x

(3)(x)

3(x)

3×x

(3) × (x)

x3

(3)x

3


Note: Since order matters with subtractions and divisions, variables and constants may appear in any place, depending on how they are being used.

F A Coefficient is a constant that is multiplied to the left of a variable or series of variables.

EXAMPLE 4: Y The coefficient of 2x is 2. Y The coefficient of

is –5.

Y The coefficient of x is 1. Y The coefficient of –y is –1. Y The coefficient of

is

.

F An Algebraic Expression is any combination of variables and constants with mathematical operations (addition, subtraction, multiplication, division, roots, and powers). Neither the equals sign (=) nor inequality symbols ( ,

,

, „, …) are part of an expression.

EXAMPLE 5: The following are algebraic expressions:

The following are not algebraic expressions: <It contains an equals sign> <It is an inequality>

F A Term is any part of an expression that is separated by plus signs (+). 4


EXAMPLE 6: List the terms of SOLUTION: The terms are:

L

. , 5x, and 7.

A subtraction can always be rewritten as “adding the opposite.” So, if you are given, for example, the following expression: It can be rewritten as: The terms are:

, –4x, and –1.

F Like Terms (or Similar Terms) are terms that have the same variables raised to the same powers, but may have different coefficients. EXAMPLE 7: Y 3x and –7x are like terms. Y

,

and

are like terms.

Y 5 and 1 are like terms. <Constants are considered to be like terms> Y

and

are like terms.

Y 6w and 8y are not like terms. <They do not have the same variables> Y 4x and

are not like terms. <The x’s are raised to different powers>

F Terms that are not like terms are called Unlike Terms. EXAMPLE 8: Identify the like terms in the following expression:

SOLUTION:

,

, and

are like terms.

–x and 7x are like terms. 4 and –5 are like terms. 5


COMBINING LIKE TERMS Before formally learning how to combine like terms together, let’s combine some fruit!

+ Three Apples

plus

= Five Apples

is equal to

Eight Apples

If we let the variable “a” represent the word “apple,” and use numbers, we get the following:

3a + 5a = 8a

Combining apples is the same as combining like terms. 3 of one variable plus 5 of the same variable gives us 8 of the variable. The result only changes in the coefficient (the 8), not the variable (the a). Now, I wish to remove two oranges from a group of oranges:

– Six Oranges

minus

= Two Oranges

is equal to

Four Oranges

If we let the variable “r” represent the word “orange,” <It is not a good idea to use the letter “O,” since it would be confused with the number zero> and use numbers, we get the following:

6r – 2r = 4r 6


Finally, I want to put both of the previous scenarios together:

+

+

=

+

Symbolically, I get:

3a + 6r + 5a – 2r = 8a + 4r To combine the above, I was only able to add apples with apples and subtract oranges from other oranges. The apples are “like” fruit and the oranges are “like” fruit. I just can’t put the apples with the oranges. To add or subtract the symbolic representations of our fruit, all what was done was to add or subtract the coefficients (the numbers). Nothing was done to change the variables (the a or the r).

F To Combine Like Terms just add (or subtract) their coefficients and attach the variable (or variables) at the end of the term. This is also called Collecting Like Terms or Simplifying An Expression. “Officially,” the combining of the terms is really using the Right Distributive Property. With the “apples,” it looks like:

3a + 5a = (3 + 5)a = 8a With the “oranges:”

6r – 2r = (6 – 2)r = 4r Personally, I think this is too complicated. (I mention it since many textbooks do). It’s easier to think of this as combining “apples with apples” and “oranges with oranges.” Often, it may be necessary to use various properties of numbers to help in the combining of like terms. This is used when you need to “show work” on homework or on a test. Using these properties can be a bit tedious, so I 7


will also show you how to work faster, but you are not showing all the steps in combining the like terms.

.

Properties of Numbers are covered in the FREE Lesson: Basic Math: Lesson 4. Go to my web site to get a copy of it.

EXAMPLE 9: Simplify the following algebraic expression by combining like terms:

SOLUTION 1 (The slow, tedious way): Y Use the Commutative Property to swap the places of the 3y and the 4x:

Y Use the Associative Property to group together the “x” terms and to group the “y” terms: Y Use the Right Distributive Property to place the variables to the right of the grouped terms:

Y Add the coefficients together for the final, simplified form:

SOLUTION 2 (The faster way): This is more clearly shown in the video version of this Lesson. Y Underline the “x” terms, then just add (or subtract) the coefficients. The “x” stays the same <some students think it should be “squared.” Don’t do this!> 2x + 3y + 4x + 5y 6x 8


Y I “cross out” the x terms, since I am done with them, then I “double underline” the y terms, and then add those terms together: 2x + 3y + 4x + 5y 6x + 8y

<Cross out the y terms too, if you wish>

You can do other things besides “underlining.” Try… Y Circling the terms: 2x Y Boxing them: 2x Y Or use “squiggles:” 2x It is recommended to cross out the terms once combining them, especially if dealing with a large number of terms, since you may accidentally add them a second time, or forget about a term.

EXAMPLE 10: Simplify:

SOLUTION: (Fast way) <Again, it will seem “smoother” in the video> Y Combine the first set of terms:

Y Cross out the “used” terms, then combine the next set of terms:

9


Y Cross out the next set of â&#x20AC;&#x153;usedâ&#x20AC;? terms, then combine the last set of terms:

If this were a test question, this is how it would look on paper:

=

Notice that only the coefficients change, the variables NEVER change when combining like terms!

If the algebraic expression has grouping symbols, use the Distributive Property to remove them first, then combine like terms.

EXAMPLE 11: Simplify the following algebraic expression:

SOLUTION: Y Distribute the 8 first:

Y Now combine like terms:

You may also write the answer as: 10


EXAMPLE 12: Combine like terms:

SOLUTION: 9A and

are unlike terms, so are 8B and , so there are NO like terms to combine; therefore the answer is just itself: <Yes, this is a trick question>

THE OPPOSITE (OR NEGATION) OF AN EXPRESSION When finding the opposite or negation of an expression, you can replace the negative sign with “–1,” then apply the Distributive Law. This changes all the signs of the terms in the expression.

EXAMPLE 13: Simplify: SOLUTION: Y Change the negation sign on the far left into a –1:

Y Distribute the –1 to the terms in the expression:

Subtraction is “adding the opposite”

The “fast way” of finding the opposite (or negation) of an algebraic expression is to just change all the signs:

11


EVALUATING EXPRESSIONS F Evaluating An Expression means replace (substitute) a variable (or variables) with numerical values, then simplify until there is only a single number.

EXAMPLE 14:

Evaluate the expression: x+y when x = 3 and y = 2

SOLUTION: Y Replace the x with the number 3, and replace the y with the number 2:

3+2

Y Perform the addition to finish the simplification: 3+2=5 The answer is 5.

HOT TIP!

(

When substituting numbers in place of variables, use parentheses first. This is especially important when dealing with negative numbers. If you realize the left parenthesis does not have a negative (minus) sign or other numbers to the left of it, then you probably don’t need to use parentheses. When in doubt… USE PARENTHESES! The Texas Instruments® graphing calculators are

capable of evaluating expressions. They have built in variables in the letters A through Z plus the Greek letter theta: U. They are the letters just above and to the right of most of the keys on the calculator. They will be represented as green, non-italicized capital letters (just as they are on the TI-84 calculator) in these Notes. There are also a few constants built into the calculators: p, e, and i. They will be represented in blue in these Notes. When referred as calculator keystrokes, they will be as follows: y<p=, y<e=, and y<i=. 12


To assign a numerical value to a variable, the value must be stored into the variable, using the key. For example, if you wish to store the value of 3 into the X variable, do the following: What to do: M 3 X <Press the again to get this X>

HOT TIP!

On the Calculator Screen: key

Since the X variable is used quite often, it may also be accessed by pressing the key (you donâ&#x20AC;&#x2122;t need to press the key first, saving a keystroke). Try using the key, then press :

What to do:

F

On the Calculator Screen:

If, when pressing the

key, another letter

appeared on the calculator screen (T, U or n), then do the following:

What to do: M z <Press the Down Arrow Key 3 times so that is blinking>

On the Calculator Screen: On the TI-83 or TI-83+ screen:

M On the TI-84+ screen:

13


Your screen may not look exactly like the above one, but make sure the “ ” is blinking, then press . This will “highlight” it and make the X appear when is pressed. Now go back and try to store a value of 3 into the X again. OK. Let’s store a 2 into the Y variable: What to do: M 2 Y <Press the again to get the Y>

On the Calculator Screen: key

Note: The arrow “è” means the value on its left is stored to the variable on its right. Finally, let’s evaluate the expression: x + y: What to do: M

On the Calculator Screen: Y

The answer is: 5

=

The TI graphing calculators use the X, Y, R, T, and U variables when graphing, so do not use them to store values if you are also performing graphing operations.

EXAMPLE 15: Evaluate the expressions below when x = 2 and y = –3: Y x – 2y

SOLUTION: Use parentheses for the y variable only, since there is

nothing directly to the left of the x, it does not need parentheses in its place. 2 – 2(–3) 14


Use the Order of Operations to simplify.

.

See “Basic Math: Lesson 3” for a detailed review of Order of Operations. 2 – 2(–3) = 2 + 6 = 8

The answer is 8. Y

SOLUTION: Use parentheses in place of the variables.

Now place the numerical values inside the parentheses:

Perform the Order of Operations to simplify to a single number:

The answer is 10. Y

SOLUTION: For this example, it is vital to use parentheses for the y variable. Here is the correct substitution:

=9–2=7 If parentheses were not used, this would be the <incorrect> answer:

15


(

Evaluate the three previous expressions when x = 2 and y = –3:

What to do: <Store the values:> M

On the Calculator Screen:

Y Y x – 2y <Enter the first expression:> Y The answer is 8. Y <Enter the second expression:> M Y The answer is 10. Y <Enter the third expression:> M Y The answer is 7.

EXAMPLE 16: Evaluate the expression:

When x = –1, y = 4, and z = 5 16


SOLUTION: Replace the variables with parentheses:

Place the numerical values inside the parentheses. Be careful you put correct numbers in the correct sets of parentheses:

The parentheses here are not really needed, but are used for consistency. Now, use the Order of Operations to simplify the expression:

The answer is 24.

(

Evaluate the expression When x = â&#x20AC;&#x201C;1, y = 4, and z = 5

What to do: <Store the values:> M

On the Calculator Screen:

Y Z <Press

to get Z> 17


<Use extra parentheses to surround the numerator and the denominator> <Enter the expression:> M Z Y

Z

The answer is 24.

ENGLISH PHRASES TO ALGEBRAIC EXPRESSIONS One of the most useful aspects of algebra is the ability to translate a phrase or sentence from a word problem into an equivalent algebraic expression or equation, then use this to solve the word problem. This part of the Lesson is an introduction of how to do this translation. In other Lessons, I will present other, more advance methods to change these English phrases into something we can manipulate mathematically. We will only translate here, not actually solve any word problems. The key to translating successfully English phrases into algebraic expressions is to recognize which words correspond to which mathematical operation. Also, the placement of numbers or variables may be important. When reading the phrases, try to imagine “placeholders” or sets of empty parentheses being used in the problem. Later, if you feel the parentheses are not needed, just remove them. I will show the “placeholders” as shaded boxes: When you work out the problems, just use parentheses or leave blank gaps. 18


L

If ambiguous words are used instead of explicitly defined variables, then you can use the variable of your choice. Try to pick them to be related to the words used. Example: if the word “number” is used in a sentence or phrase, use the variable “n.” If “distance” is mentioned, try using a “d.” If “time” is of concern, use a “t.” Of course, you are welcome to use the old tried and true variable: x.

F ADDITION F Y PHRASE: “The sum of… ” When using “sum of,” visualize the following:

+ = EXAMPLE PHRASE: The sum of a number and 2. When reading “the sum of,” use the “+” with the placeholders (or parentheses):

+ or

( )+( ) “A number” means the variable. Since none was specified, you can use any you wish. I will use “n.” Place this to the left of the plus sign and the number 2 to the right of the plus sign:

n + 2 or

(n)+(2) So the answer is: n + 2. <If you realize the parentheses are not needed, just remove them> 19


Y PHRASE: “ …more than… ” When using “more than,” place the first item to the rightÌ of the plus sign. The second item is placed to the Ëleft. They “swap” places. Of course, “officially,” it doesn’t matter how items are placed in a sum, but some math instructors would penalize students for not doing this “swap.” = EXAMPLE PHRASE: 3 more than a number. When reading “3 more than,” do the following:

+ 3 <the 3 goes to the right> Select a variable to place in the blank spot:

n + 3 The answer is n + 3. Y WORD: “ … plus… ” This is pretty straightforward to use. Place items in the order presented. = EXAMPLE PHRASE: 5 plus some number.

5 + 5 + n <Select variable of your choice> The answer is: 5 + n. Y PHRASE: “ …increased by… ” Use in the same way as previous phrase. = EXAMPLE PHRASE: A number increased by 1.

n + n + 1 Answer is: n + 1. 20


F SUBTRACTION F Order is VERY IMPORTANT when translating phrases that represent subtractions. This is probably the toughest section on creating algebraic expressions. Here are the “non-swapping” phrases (the order mentioned in phrase is the SAME as that listed in the mathematical expression): Y PHRASE: “The difference of… ” = EXAMPLE PHRASE: The difference of x and y. When reading “difference of,” write down:

– Since x is mentioned first, it is placed first. The y is placed second:

x – y The answer is: x – y. Y PHRASE: “ …decreased by… ” Treat this like the previous phrase. = EXAMPLE PHRASE: 9 is decreased by w.

– 9 – w The answer is: 9 – w. Y PHRASES: “ …minus… ” or “ …reduced by… ” These are translated in the same way as the previous phrases. = EXAMPLE PHRASES: ✜ A number minus 3 " n – 3. 21


✜ A number is reduced by 8 " n – 8.

= In the next two phrases, the values stated “swap” places. Y PHRASE: “ …less than… ” = EXAMPLE PHRASE: 10 less than a number. The 10 is placed to the rightÌ of the minus sign:

– 10 Place the variable in the blank spot:

n – 10 The answer is: n – 10.

= The above phrase is the one most students get wrong on tests. 10 less than a number is: n – 10 NOT: 10 – n Y PHRASE: “ …subtracted from… ” = EXAMPLE PHRASE: 7 is subtracted from y. When reading “subtracted from,” place the first value to the rightÌ of the minus sign:

– 7 Place the y in the first spot:

y – The answer is: y – 7.

22

7


F MULTIPLICATION F When setting up multiplications, use “implied multiplication” (numbers are just next to the variables) when possible. Using parentheses for clarity is highly recommended. Avoid using the raised dot ( ) and the times sign ( × ).

Basic Strategy:

I will usually start off using parentheses in the first step, then rewrite the expression in a simpler form, if possible.

Order is not important with multiplication, so just list the items in the order presented in the phrase. Y PHRASE: “The product of… ” = EXAMPLE PHRASE: The product of 8 and a number. When reading “product of,” set up parentheses:

(

)(

)

Fill them in with the values mentioned (make up a variable, if it is not explicitly stated):

(8)(n)

Rewrite it as an implied multiplication, for the final answer: 8n With practice, you should be able to go from step to step . Don’t do this if the phrase contains more elaborate items (more on this later). Now, had the phrase been: “The product of a number and 8.” You can translate this as follows:

(

)(

)

(n)(8) 23


At this point, you need to be careful. Some math instructors do not want you to “swap” places; so leave the answer as (n)(8). The better answer would be to swap the order or the factors, so that the constant is to the left of the variable: (n)(8) ⇔ 8n 8n is using implied multiplication and is a more simplified expression. Y PHRASES: “ …multiplied by… ” or “ …times… ” These are straightforward translations. = EXAMPLE PHRASES: ✜ A number multiplied by four → (n)(4) <which simplifies to:> → 4n. ✜ Seven times a number → (7)(n) → 7n. Y PHRASES THAT MAKE THINGS LARGER: Phrase: “Twice…” “Double…” “Triple…” “Quadruple…”

Example: Twice a number. Double the rate, r. Triple the score. Quadruple the calories.

Translation: 2( ) 2( ) 3( ) 4( )

With Variables: 2n 2r 3s 4c

Other likely phrases include: “thrice,” “quintuple,” “sextuple,” etc. Y WORD: “ …of… ” The word “of” may be used to mean multiplication when combined with fractions or percentages. = EXAMPLE PHRASES: ✜ Half of a number → ✜ Two-thirds of z → 24


L

When translating percentages to algebraic expressions, almost always you will need to first convert them to a decimal. Here is an example:

✜ Twenty percent of a price. “Twenty percent” → 20 20

→ 0.20 <You may also just use: .2 >

“of” means multiply by what follows it. In this case the price of something. We will represent this with a variable, say, p. The final expression is 0.20p or .2p.

F DIVISION F There are many ways of mathematically expressing a division. Here are the most common ways of showing “x divided by y:” x:y My recommendation is to use the horizontal fraction bar (––) for translating purposes. It is the least ambiguous one, especially when dealing with more complicated expressions. Usually, you will use colons ( : ) when dealing with ratios (there are exceptions to this).

Basic Strategy:

Start off by writing a horizontal fraction bar with “placeholders:” parentheses:

or, if rather complicated, with .

Y PHRASE: “The quotient of _______ and _______” This is probably the most typical phrase for division. The first item is placed in the numerator (top) and the second item, the one mentioned after the word “and,” is placed in the denominator (bottom). 25


= EXAMPLE PHRASE: The quotient of a number and five. When reading “quotient of,” imagine:

The first item, “the number,” is the numerator. In this case, let’s use the ever popular n as our variable:

The item after the word “and” (the five) is the denominator:

The answer is: Y PHRASES: “Divide ______ by ______” or “ …divided by… ” Again, the first item is placed in the numerator and the second item, the one mentioned after the word “by,” is placed in the denominator. = EXAMPLE PHRASES: ✜ Divide three by a number →

✜ A number divided by four →

Y PHRASE: “The ratio of ______ to ______” Use a colon instead of a fraction bar: : The first item is placed on left placeholder, the second on other side. 26


= EXAMPLE PHRASE: The ratio of boys to girls. Let “b” represent the boys, and let “g” be the girls: →b:g

:

Note: If you are doing more than just translating phrases with “ratio” (perhaps also solving equations), then it may be better to use a horizontal fraction bar instead of a colon. Y WORD: “ …per… ” “Per” is usually used to divide units of measurement. A slanted fraction bar may be used in this case. Abbreviations of the units may be used instead of variables, if applicable. = EXAMPLE PHRASES: ✜ Miles per gallon. → ✜ Feet per second. → ✜ Hot dogs per minute →

<J>

F ADDITIONAL PHRASES F Y PHRASES: “ …squared”, “The square of… ”, “ …cubed”, “The cube of… ” When reading “square,” imagine a “placeholder” or set of parentheses being raised to the second power: 2

or

The item(s) mentioned is/are placed inside.

27


Likewise, when reading “cube,” imagine a “placeholder” or set of parentheses being raised to the third power: 3

or

= EXAMPLE PHRASES: ✜ A number squared. → ✜ The square of a number. → ✜ A number cubed. → ✜ The cube of a number. → Y PHRASE: “What percent… ” When reading the phrase: “what percent,” you write a variable (usually x) and divide it by 100:

This will be seen in more detail in other lessons. Y PHRASES THAT TYPICALLY APPEAR ON STANDARDIZED TESTS: = “CONSECUTIVE INTEGERS” Use the following set up: n, n + 1, n + 2, n + 3, n + 4, etc. <Depending on how many are asked for> = “EVEN NUMBER” Use: 2n to represent an even number. = “ODD NUMBER” Use: 2n + 1 or 2n – 1 to represent an odd number. 28


= “CONSECUTIVE EVEN (OR ODD) NUMBERS” Use the following set up for EITHER situation: n, n + 2, n + 4, n + 6, n + 8, etc. <Depending on how many are asked for> Note: If you are translating a phrase into an equation to later solve, you don’t need to start using “2n” (for evens) or “2n + 1” (for odds) when creating phrases using consecutive even or odd numbers. If you were just stating an expression (and NOT solving), then the following would do: • Consecutive Even Numbers: 2n, 2n + 2, 2n + 4, 2n + 6, etc. • Consecutive Odd Numbers: 2n + 1, 2n + 3, 2n + 5, 2n + 7, etc. Y OTHER MATHEMATICAL PHRASES: Occasionally, other mathematical phrases may be used, such as “absolute value,” “square root,” “the reciprocal of…,” etc. Most can be treated as grouping symbols, like parentheses. Reading the phrase carefully, “filling in” the expression in steps, and using the placeholders will help you with more complicated phrases. A Final Observation… I have tried to provide you with the most common phrases used in word problems. This is not, by far, a complete list. Using common sense, and seeing how words are used in a problem, will help you in cases when you see something not mentioned here. There may also be “stealth operations” within a phrase. Most of these are really just multiplications, but not always. You may need to convert units (using multiplication and/or division) or use special formulas to help you set up an expression.

29


F MORE EXAMPLES F Let’s practice with more complex phrases, paying special attention on how small changes in wording can give us different algebraic expressions.

HOT TIP!

A comma ( , ) used in a phrase provides helpful separation of terms or groupings of smaller expressions.

EXAMPLE 17: Translate the following phrase into an algebraic expression: “The sum of twice a number and three.”

SOLUTION:

“The sum of” +

“Twice a number”

2n +

“and three.”

2n + 3

EXAMPLE 18: Translate the following phrase into an algebraic expression: “Twice the sum of a number and three.”

SOLUTION: “The sum of” “a number” “and three.”

“Twice” 2( 2(

) +

2(n +

) )

2(n + 3) EXAMPLE 19: Translate the following phrase into an algebraic expression: “Eight more than six times a number.”

30


SOLUTION:

“Eight more than” <When using “more than,” place the first item to the end (the right hand side)> +8

“six times a number”

6n + 8 EXAMPLE 20: Translate the following phrase into an algebraic expression: “Triple the difference of one and a number.”

SOLUTION:

“Triple” 3(

“The difference of” “one” “and a number.”

3(

) –

)

3(1 –

)

3(1 – n)

EXAMPLE 21: Translate the following phrase into an algebraic expression: “Nine less than the quotient of x and two.”

SOLUTION: “the quotient of”

“Nine less than” –9

Put the placeholders for division in the place of the original shaded area of the subtraction: –9 Continuing the previous step: “the quotient of x and two”

31


EXAMPLE 22: Translate the following phrase into an algebraic expression: “The quotient of nine less than x and two.”

SOLUTION:

“The quotient of ___ and ___”

Everything before the word “and” belongs in the numerator: “nine less than”

“x”

Whatever is past the word “and” goes in the denominator: “and two.”

EXAMPLE 23: Translate the following phrase into an algebraic expression: “Double the sum of a number and seven, all divided by the square of the number.”

SOLUTION:

“Double”

“The sum of” “a number and seven”

2( 2(

) +

2(n + 7) 32

)


The comma provides a separation of items. The “all divided by…” phrase means divide the expression we just got by what follows the word “by”:

“the square of”

“the number.” <Use the same variable as before>

EXAMPLE 24: Write algebraic expressions for these similarly worded phrases: : “The sum of the squares of two numbers.” : “Square the sum of two numbers.”

SOLUTION FOR

:

“The sum of”

+

“the squares of two numbers.” This means the numbers are both being individually squared. Since no specific variables are mentioned, I will use x and y:

SOLUTION FOR

:

“Square" <Parentheses used since I am anticipating something complicated will be placed inside them> 33


“The sum of”

“two numbers.”

EXAMPLE 25: Translate the following phrase into an algebraic expression: “The product of three consecutive numbers.”

SOLUTION:

“the product of” usually implies two factors, but as you read the phrase, you should realize there are three factors, so an extra set of parentheses will be used:

“three consecutive numbers” can be represented as: n, n + 1, and n + 2. Placing them within the parentheses, we get:

EXAMPLE 26: Translate the following phrase into an algebraic expression: “The quotient of two consecutive numbers.”

SOLUTION:

“The quotient of”

“two consecutive numbers” can be represented as: n and n + 1. Place the first over the second:

EXAMPLE 27: Translate the following phrase into an algebraic expression: “A third of half a number.” 34


SOLUTION:

“A third of”

“half a number.”

EXAMPLE 28: Translate and simplify, if possible, the following phrase into an algebraic expression: “The value, in dollars, of q quarters.”

SOLUTION:

The value of one quarter, in dollars, is $0.25.

There are q quarters, so we want an expression for 0.25 times q quarters: Simplified:

0.25q

Note: Units are usually not given in the expression. EXAMPLE 29: Translate the following phrase into an algebraic expression: “Two-fifths of the difference between nine and the cube of a number.”

SOLUTION:

“Two-fifths of”

“the difference between ____ and ____” When using “difference,” the items are placed in the order listed.

35


“nine”

“and the cube of”

“a number.”

EXAMPLE 30: Translate the following phrase into an algebraic expression: “The product of the squares of two consecutive even numbers.”

SOLUTION:

“The product of”

“the squares of two…” means both items are squared individually”

“two consecutive even numbers.” This can be written in two ways, depending on how you define the variable. If n = an even number, then the two consecutive numbers are n and n + 2. The expression then becomes:

However, if 2n = an even number, then the two consecutive numbers are 2n and 2n + 2. The expression is now:

36


For solving purposes, the first one is better, but the second one is clearer that we are dealing with even numbers. EXAMPLE 31: If a loan for a new house is L, what expression would represent a 15 down payment of this loan?

SOLUTION: First try to make this problem a little less “wordy.” • The loan is L. • We want “15 of loan L.” “15

of” is rewritten as:

The loan is L: Simplified:

0.15(

)

0.15(L) 0.15L

EXAMPLE 32: A car can travel m miles using g gallons of gas. What is an expression representing the number of miles per gallon used?

SOLUTION:

“per” means division:

“miles per gallon”

IT’S QUIZ TIME!

37


LESSON QUIZ 1 For the following algebraic expression: 3x + 5 Y What is the variable? Answer: ____ Y What is the constant? Answer: ____ Y What is the coefficient? Answer: ____

2 What is the coefficient of the following expressions? Y –x Answer: ____ Y (y)(4) Answer: ____ Y

Answer: ____

Y 2πr Answer: ____

3 Identify if the following are expressions, equations or inequalities: Y

Answer: ______________

Y ax + b Answer: ______________ Y xy Y

1 Answer: ______________ Answer: ______________

4 List the terms of: The terms are: __________________________________ 38


5 Identify the like terms in the following algebraic expression:

Answer:

6 Simplify the following algebraic expressions by combining like terms: Y 3x – 2y – 4x + 3y Answer: Y Answer:

7 Simplify: Answer:

8 Evaluate the expression: 3x – 2y when x = 4 and y = –5 Answer:

9 Evaluate the expression:

when x = 3 and y = 2

Answer:

bl Translate the following phrases into algebraic expressions. Use n if no variable is mentioned. Y Three less than twice a number. Answer: 39


Y Triple the difference of y and z. Answer: Y Four more than the product of eight and a number. Answer: Y The quotient of seven and the absolute value of a number. Answer: Y The absolute value of the quotient of seven and a number. Answer: Y Two less than the product of four consecutive numbers. Answer: Y A number incremented by one is multiplied by the same number decreased by two, all divided by the number. Answer:

Y The price p of a car is discounted by 10 Answer:

ANSWERS ON NEXT PAGEâ&#x20AC;Ś

40


ANSWERS

1 For the following algebraic expression: 3x + 5 Y What is the variable? Answer: x Y What is the constant? Answer: 5 Y What is the coefficient? Answer: 3

2 What is the coefficient of the following expressions? Y –x Answer: –1 Y (y)(4) Answer: 4 Y

Answer:

It is not 1, since (y)(4) = 4y

#1

Y 2πr Answer: 2π

3 Identify if the following are expressions, equations or inequalities: Answer: Equation

Y

Y ax + b Answer: Expression Y xy Y

1 Answer: Inequality Answer: Expression

4 List the terms of: 3

2

The terms are: –2x , 5x , –x and 8 41


5 Identify the like terms in the following algebraic expression:

Answer:

–3x and 7x are like terms. 2 2 2 4y , y , and –πy , are like terms.

6 Simplify the following algebraic expressions by combining like terms: Y 3x – 2y – 4x + 3y Answer: –x + y (or y – x) Y 2

2

Answer: 6xy – 4y + 5x

7 Simplify: 2

Answer: 2x – 5y + 8y

8 Evaluate the expression: 3x – 2y when x = 4 and y = –5 Answer: 22

9 Evaluate the expression:

when x = 3 and y = 2

Answer: –10

bl Translate the following phrases into algebraic expressions. Use n if no variable is mentioned. Y Three less than twice a number. Answer: 2n – 3 42


Y Triple the difference of y and z. Answer: 3(y – z) Y Four more than the product of eight and a number. Answer: 8n + 4 Y The quotient of seven and the absolute value of a number. Answer:

Y The absolute value of the quotient of seven and a number. Answer: Y Two less than the product of four consecutive numbers. Answer:

(n)(n + 1)(n + 2)(n + 3) – 2 or n(n + 1)(n + 2)(n + 3) – 2

Y A number incremented by one is multiplied by the same number decreased by two, all divided by the number. Answer: Y The price p of a car is discounted by 10 Answer: “discounted by” means subtract, so we need to subtract 10 the price from the original price. This can be represented by: p – 0.10p or p – .1p or .9p

END OF LESSON 43

of

algebra1  

intro to algebra

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