Chapter 11. Algebraic Expressions and Equations
11.1. Objectives*
After completing this chapter, you should
Algebraic Expressions (Section 11.2)
be able to recognize an algebraic expression
be able to distinguish between terms and factors
understand the meaning and function of coefficients
be able to perform numerical evaluation
Combining Like Terms Using Addition and Subtraction (Section 11.3)
be able to combine like terms in an algebraic expression
Solving Equations of the Form 
and 
(Section 11.4)
understand the meaning and function of an equation
understand what is meant by the solution to an equation
be able to solve equations of the form
and
Solving Equations of the Form 
and 
(Section 11.5)
be familiar with the multiplication/division property of equality
be able to solve equations of the form
and
be able to use combined techniques to solve equations
Applications I: Translating Words to Mathematical Symbols (Section 11.6)
be able to translate phrases and statements to mathematical expressions and equations
Applications II: Solving Problems (Section 11.7)
be more familiar with the five-step method for solving applied problems
be able to use the five-step method to solve number problems and geometry problems
11.2. Algebraic Expressions*
Section Overview
Algebraic Expressions
Terms and Factors
Coefficients
Numerical Evaluation
Algebraic Expressions
In arithmetic, a numerical expression results when numbers are connected by arithmetic operation signs (+, -, ⋅ , ÷). For example,
,
,
, and
are numerical expressions.
In algebra, letters are used to represent numbers, and an algebraic expression results when an arithmetic operation sign associates a letter with a number or a letter with a letter. For example,
,
,
,
, and
are algebraic expressions.
Numerical expressions and algebraic expressions are often referred to simply as expressions.
Terms and Factors
In algebra, it is extremely important to be able to distinguish between terms and factors.
Terms are parts of sums and are therefore connected by + signs. Factors are parts of products and are therefore separated by ⋅ signs.
Note
While making the distinction between sums and products, we must remember that subtraction and division are functions of these operations.
In some expressions it will appear that terms are separated by minus signs. We must keep in mind that subtraction is addition of the opposite, that is,
In some expressions it will appear that factors are separated by division signs. We must keep in mind that
Sample Set A
State the number of terms in each expression and name them.
Example 11.1.
. In this expression, x and 4 are connected by a "+" sign. Therefore, they are terms. This expression consists of two terms.
Example 11.2.
. The expression
can be expressed as
. We can now see that this expression consists of the two terms
and
.
Rather than rewriting the expression when a subtraction occurs, we can identify terms more quickly by associating the + or - sign with the individual quantity.
Example 11.3.
. Associating the sign with the individual quantities, we see that this expression consists of the four terms
, 7,
,
.
Example 11.4.
. This expression consists of the two terms,
and
. Notice that the term
is composed of the two factors 5 and
. The term
is composed of the two factors
and
.
Example 11.5.
. This expression consists of one term. Notice that
can be expressed as
or
(indicating the connecting signs of arithmetic). Note that no operation sign is necessary for multiplication.
Practice Set A
Specify the terms in each expression.
Coefficients
We know that multiplication is a description of repeated addition. For example,
describes
Suppose some quantity is represented by the letter
. The multiplication
describes
. It is now easy to see that
specifies 5 of the quantities represented by
. In the expression
, 5 is called the numerical coefficient, or more simply, the coefficient of
.
The coefficient of a quantity records how many of that quantity there are.
Since constants alone do not record the number of some quantity, they are not usually considered as numerical coefficients. For example, in the expression
, the coefficient of
is 7. (There are 7 x's.)
is 2. (There are 2 y 's.)
is
. (There are
z 's.)
The constant 12 is not considered a numerical coefficient.
When the numerical coefficient of a variable is 1, we write only the variable and not the coefficient. For example, we write
rather than
. It is clear just by looking at
that there is only one.
Numerical Evaluation
We know that a variable represents an unknown quantity. Therefore, any expression that contains a variable represents an unknown quantity. For example, if the value of
is unknown, then the value of
is unknown. The value of
depends on the value of
.
Numerical evaluation is the process of determining the numerical value of an algebraic expression by replacing the variables in the expression with specified numbers.
Sample Set B
Find the value of each expression.
Example 11.6.
, if
and
Replace x with –4 and y with 2.
Thus, when 
and
,
.
Example 11.7.
, if 
and 
= 
.
Replace a with 6 and b with –3.
Thus, when a = 6 and b = –3,
.
Example 11.8.
, if
and
Replace 
with –5 and 
with –1.
Thus, when 
and 
,
.
Example 11.9.
, if
Replace x with 4.
Thus, when 
,
.
Example 11.10.
, if
Replace x with 3.
Example 11.11.
, if
.
Replace x with 3.
The exponent is connected to –3, not 3 as in the problem above.
Practice Set B
Find the value of each expression.
Exercises
Exercise 11.2.14. (Go to Solution)
In an algebraic expression, terms are separated by _______________ signs and factors are separated by _______________ signs.
For the following 8 problems, specify each term.
Exercise 11.2.15.
Exercise 11.2.17.
Exercise 11.2.19.
Exercise 11.2.21.
Exercise 11.2.23.
What is the function of a numerical coefficient?
Exercise 11.2.25.
Write 1s in a simpler way.
Exercise 11.2.27.
In the expression –7c, how many c’s are indicated?
Find the value of each expression.
Exercise 11.2.29.
, if
and
Exercise 11.2.31.
, if
,
,
, and
Exercise 11.2.33.
if
and
Exercise 11.2.35.
, if
and
Exercise 11.2.37.
, if
,
, and
Exercise 11.2.39.
, if
,
,
, and
Exercise 11.2.41.
, if
and
Exercise 11.2.43.
, if
and
Exercise 11.2.45.
, if
Exercise 11.2.47.
if
Exercise 11.2.49.
if
Exercise 11.2.51.
if
Exercises for Review
Solutions to Exercises
11.3. Combining Like Terms Using Addition and Subtraction*
Section Overview
Combining Like Terms
Combining Like Terms
From our examination of terms in Section 11.2, we know that like terms are terms in which the variable parts are identical. Like terms is an appropriate name since terms with identical variable parts and different numerical coefficients represent different amounts of the same quantity. When we are dealing with quantities of the same type, we may combine them using addition and subtraction.
An algebraic expression may be simplified by combining like terms.
This concept is illustrated in the following examples.
Eight and 5 of the same type give 13 of that type. We have combined quantities of the same type.
Eight and 5 of the same type give 13 of that type. Thus, we have 13 of one type and 3 of another type. We have combined only quantities of the same type.
Suppose we let the letter
represent "record." Then,
. The terms
and
are like terms. So, 8 and 5 of the same type give 13 of that type. We have combined like terms.
- Suppose we let the letter
represent "record" and 
represent "tape." Then,
We have combined only the like terms.
After observing the problems in these examples, we can suggest a method for simplifying an algebraic expression by combining like terms.
Like terms may be combined by adding or subtracting their coefficients and affixing the result to the common variable.
Sample Set A
Simplify each expression by combining like terms.
Example 11.12.
. All three terms are alike. Combine their coefficients and affix this result to 
: 
.
Thus,
.
Example 11.13.
The terms
and
are like terms. Combine their coefficients:
.
Thus,
.
Example 11.14.
The like terms are
Thus,
Example 11.15.
. The like terms are
Thus,
.
Practice Set A
Simplify each expression by combining like terms.
Exercises
Simplify each expression by combining like terms.
Exercise 11.3.7.
Exercise 11.3.9.
Exercise 11.3.11.
Exercise 11.3.13.
Exercise 11.3.15.
Exercise 11.3.17.
Exercise 11.3.19.
Exercise 11.3.21.
Exercise 11.3.23.
Exercise 11.3.25.
Exercises for Review
Solutions to Exercises
11.4. Solving Equations of the Form x+a=b and x-a=b*
Section Overview
Equations
Solutions and Equivalent Equations
Solving Equations
Equations
An equation is a statement that two algebraic expressions are equal.
The following are examples of equations:
Notice that
,
, and
are not equations. They are expressions. They are not equations because there is no statement that each of these expressions is equal to another expression.
Solutions and Equivalent Equations
The truth of some equations is conditional upon the value chosen for the variable. Such equations are called conditional equations. There are two additional types of equations. They are examined in courses in algebra, so we will not consider them now.
The set of values that, when substituted for the variables, make the equation true, are called the solutions of the equation. An equation has been solved when all its solutions have been found.
Sample Set A
Example 11.16.
Verify that 3 is a solution to
.
When
,
Example 11.17.
Verify that 
is a solution to
When
,
Example 11.18.
Verify that 5 is not a solution to
.
When
,
Example 11.19.
Verify that -2 is a solution to
.
When
,
Practice Set A
Some equations have precisely the same collection of solutions. Such equations are called equivalent equations. For example,
,
, and
are all equivalent equations since the only solution to each is
. (Can you verify this?)
Solving Equations
We know that the equal sign of an equation indicates that the number represented by the expression on the left side is the same as the number represented by the expression on the right side.
| This number | is the same as | this number |
| ↓ | ↓ | ↓ |
| = | 4 |
| = | 11 |
| = | -1 |
From this, we can suggest the addition/subtraction property of equality. Given any equation,
We can obtain an equivalent equation by adding the same number to both sides of the equation.
We can obtain an equivalent equation by subtracting the same number from both sides of the equation.
The idea behind equation solving is to isolate the variable on one side of the equation. Signs of operation (+, -, ⋅,÷) are used to associate two numbers. For example, in the expression
, the numbers 5 and 3 are associated by addition. An association can be undone by performing the opposite operation. The addition/subtraction property of equality can be used to undo an association that is made by addition or subtraction.
Subtraction is used to undo an addition.
Addition is used to undo a subtraction.
The procedure is illustrated in the problems of id1163760715212.
Sample Set B
Use the addition/subtraction property of equality to solve each equation.
Example 11.20.
. 4 is associated with
by addition. Undo the association by subtracting 4 from both sides.
Check: When
,
becomes
The solution to
is
.
Example 11.21.
. 8 is associated with
by subtraction. Undo the association by adding 8 to both sides.
Check: When
,
becomes 
a true statement.
The solution to
is
.
Example 11.22.
. Before we use the addition/subtraction property, we should simplify as much as possible.
6 is associated with
by addition. Undo the association by subtracting 6 from both sides.
This is equivalent to
.
Check: When
,
becomes 
, a true statement.
The solution to
is
.
Example 11.23.
. Begin by simplifying the left side of the equation.
1 is associated with
by addition. Undo the association by subtracting 1 from both sides.
Check: When
,
becomes 
, a true statement.
The solution to
is
.
Example 11.24.
. In this equation, the variable appears on both sides. We need to isolate it on one side. Although we can choose either side, it will be more convenient to choose the side with the larger coefficient. Since 8 is greater than 6, we’ll isolate
on the left side.
Since
represents
, subtract
from each side.
4 is associated with
by subtraction. Undo the association by adding 4 to both sides.
Check: When
,
becomes 
a true statement.
The solution to
is
.
Example 11.25.
. -8 is associated with
by addition. Undo the by subtracting -8 from both sides. Subtracting -8 we get
. We actually add 8 to both sides.
Check: When
becomes 
, a true statement.
The solution to
is
.
Practice Set B
Exercises
For the following 10 problems, verify that each given value is a solution to the given equation.
Exercise 11.4.13.
,
Exercise 11.4.15.
,
Exercise 11.4.17.
,
Exercise 11.4.19.
,
Exercise 11.4.21.
,
Solve each equation. Be sure to check each result.
Exercise 11.4.23.
Exercise 11.4.25.
Exercise 11.4.27.
Exercise 11.4.29.
Exercise 11.4.31.
Exercise 11.4.33.
Exercise 11.4.35.
Exercise 11.4.37.
Exercise 11.4.39.
Exercise 11.4.41.
Exercise 11.4.43.
Exercise 11.4.45.
Exercise 11.4.47.
Exercise 11.4.49.
Exercise 11.4.51.
Exercises for Review
Solutions to Exercises
11.5. Solving Equations of the Form ax=b and x/a=b*
Section Overview
Multiplication/ Division Property of Equality
Combining Techniques in Equations Solving
Multiplication/ Division Property of Equality
Recall that the equal sign of an equation indicates that the number represented by the expression on the left side is the same as the number represented by the expression on the right side. From this, we can suggest the multiplication/division property of equality.
Given any equation,
We can obtain an equivalent equation by multiplying both sides of the equation by the same nonzero number, that is, if
, then
is equivalent to
We can obtain an equivalent equation by dividing both sides of the equation by the same nonzero number, that is, if
, then
is equivalent to
The multiplication/division property of equality can be used to undo an association with a number that multiplies or divides the variable.
Sample Set A
Use the multiplication / division property of equality to solve each equation.
Example 11.26.
6 is associated with y by multiplication. Undo the association by dividing both sides by 6
Check: When
becomes 
, a true statement.
The solution to
is
.
Example 11.27.
. -2 is associated with
by division. Undo the association by multiplying both sides by -2.
Check: When
,
becomes 
a true statement.
The solution to
is
Example 11.28.
. We will examine two methods for solving equations such as this one.
Method 1: Use of dividing out common factors.
7 is associated with
by division. Undo the association by multiplying both sides by 7.
Divide out the 7’s.
3 is associated with
by multiplication. Undo the association by dviding both sides by 3.
Check: When
,
becomes 
, a true statement.
The solution to
is
.
Method 2: Use of reciprocals
Recall that if the product of two numbers is 1, the numbers are reciprocals. Thus
and
are reciprocals.
Multiply both sides of the equation by
, the reciprocal of
.
Notice that we get the same solution using either method.
Example 11.29.
-8 is associated with 
by multiplication. Undo the association by dividing both sides by -8.
Check: When
,
becomes 
, a true statement.
Example 11.30.
Since 
is actually
and
, we can isolate 
by multiplying both sides of the equation by
.
Check: When
,
becomes 
The solution to
is
.
Practice Set A
Use the multiplication/division property of equality to solve each equation. Be sure to check each solution.
Combining Techniques in Equation Solving
Having examined solving equations using the addition/subtraction and the multiplication/division principles of equality, we can combine these techniques to solve more complicated equations.
When beginning to solve an equation such as
, it is helpful to know which property of equality to use first, addition/subtraction or multiplication/division. Recalling that in equation solving we are trying to isolate the variable (disassociate numbers from it), it is helpful to note the following.
To associate numbers and letters, we use the order of operations.
Multiply/divide
Add/subtract
To undo an association between numbers and letters, we use the order of operations in reverse.
Add/subtract
Multiply/divide
Sample Set B
Solve each equation. (In these example problems, we will not show the checks.)
Example 11.31.
-4 is associated with 
by subtraction. Undo the association by adding 4 to both sides.
6 is associated with 
by multiplication. Undo the association by dividing both sides by 6
Example 11.32.
3 is associated with 
by addition. Undo the association by subtracting 3 from both sides.
-8 is associated with 
by multiplication. Undo the association by dividing both sides by -8.
Example 11.33.
Begin by solving this equation by combining like terms.
Choose a side on which to isolate m. Since 7 is greater than 1, we'll isolate m on the right side.
Subtract m from both sides.
8 is associated with m by subtraction. Undo the association by adding 8 to both sides.
6 is associated with m by multiplication. Undo the association by dividing both sides by 6.
Reduce.
Notice that if we had chosen to isolate m on the left side of the equation rather than the right side, we would have proceeded as follows:
Subtract 
from both sides.
Add 6 to both sides,
Divide both sides by -6.
This is the same result as with the previous approach.
Example 11.34.
7 is associated with 
by division. Undo the association by multiplying both sides by 7.
8 is associated with 
by multiplication. Undo the association by dividing both sides by 8.
Practice Set B
Solve each equation. Be sure to check each solution.
Exercises
Solve each equation. Be sure to check each result.
Exercise 11.5.14.
Exercise 11.5.16.
Exercise 11.5.18.
Exercise 11.5.20.
Exercise 11.5.22.
Exercise 11.5.24.
Exercise 11.5.26.
Exercise 11.5.28.
Exercise 11.5.30.
Exercise 11.5.32.
Exercise 11.5.34.
Exercise 11.5.36.
Exercise 11.5.38.
Exercise 11.5.40.
Exercise 11.5.42.
Exercise 11.5.44.
Exercise 11.5.46.
Exercise 11.5.48.
Exercises for Review
Exercise 11.5.50.
Solutions to Exercises
11.6. Applications I: Translating Words to Mathematical Symbols*
Section Overview
Translating Words to Symbols
Translating Words to Symbols
Practical problems seldom, if ever, come in equation form. The job of the problem solver is to translate the problem from phrases and statements into mathematical expressions and equations, and then to solve the equations.
As problem solvers, our job is made simpler if we are able to translate verbal phrases to mathematical expressions and if we follow the five-step method of solving applied problems. To help us translate from words to symbols, we can use the following Mathematics Dictionary.
| MATHEMATICS DICTIONARY | |
| Word or Phrase | Mathematical Operation |
| Sum, sum of, added to, increased by, more than, and, plus | + |
| Difference, minus, subtracted from, decreased by, less, less than | - |
| Product, the product of, of, multiplied by, times, per | ⋅ |
| Quotient, divided by, ratio, per | ÷ |
| Equals, is equal to, is, the result is, becomes | = |
| A number, an unknown quantity, an unknown, a quantity |
 (or any symbol) |
Sample Set A
Translate each phrase or sentence into a mathematical expression or equation.
Example 11.35.
Translation:
.
Example 11.36.
Translation:
.
Example 11.37.
Translation:
.
Example 11.38.
Translation:
.
Example 11.39.
Translation:
, or
.
Example 11.40.
Translation:
.
Practice Set A
Translate each phrase or sentence into a mathematical expression or equation.
Exercise 11.6.6. (Go to Solution)
Three more than seven times a number is nine more than five times the number.
Exercise 11.6.7. (Go to Solution)
Twice a number less eight is equal to one more than three times the number.
Sample Set B
Example 11.41.
Sometimes the structure of the sentence indicates the use of grouping symbols. We’ll be alert for commas. They set off terms.
Translation:
.
Example 11.42.
Some phrases and sentences do not translate directly. We must be careful to read them properly. The word from often appears in such phrases and sentences. The word from means “a point of departure for motion.” The following translation will illustrate this use.
Translation:
.
The word from indicated the motion (subtraction) is to begin at the point of “some number.”
Example 11.43.
Ten less than some number. Notice that less than can be replaced by from.
Ten from some number.
Translation:
.
Practice Set B
Translate each phrase or sentence into a mathematical expression or equation.
Exercise 11.6.9. (Go to Solution)
A number divided by three, minus the same number multiplied by six, is one more than the number.
Exercises
Translate each phrase or sentence to a mathematical expression or equation.
Exercise 11.6.13.
Six more than an unknown number.
Exercise 11.6.15.
A number plus seven.
Exercise 11.6.17.
A number decreased by ten.
Exercise 11.6.19.
Negative nine added to a number.
Exercise 11.6.21.
A number minus the opposite of five.
Exercise 11.6.23.
A number minus the opposite of negative twelve.
Exercise 11.6.25.
Six plus five times an unknown number.
Exercise 11.6.27.
Ten times a quantity increased by two is nine.
Exercise 11.6.29.
Four times a number minus twenty-nine is eleven.
Exercise 11.6.31.
Two ninths of a number plus one fifth is forty-one.
Exercise 11.6.32. (Go to Solution)
When four thirds of a number is increased by twelve, the result is five.
Exercise 11.6.33.
When seven times a number is decreased by two times the number, the result is negative one.
Exercise 11.6.34. (Go to Solution)
When eight times a number is increased by five, the result is equal to the original number plus twenty-six.
Exercise 11.6.35.
Five more than some number is three more than four times the number.
Exercise 11.6.36. (Go to Solution)
When a number divided by six is increased by nine, the result is one.
Exercise 11.6.37.
A number is equal to itself minus three times itself.
Exercise 11.6.39.
A number divided by nine, minus five times the number, is equal to one more than the number.
Exercise 11.6.41.
When four is subtracted from some number, the result is thirty-one.
Exercise 11.6.42. (Go to Solution)
Three less than some number is equal to twice the number minus six.
Exercise 11.6.43.
Thirteen less than some number is equal to three times the number added to eight.
Exercise 11.6.44. (Go to Solution)
When twelve is subtracted from five times some number, the result is two less than the original number.
Exercise 11.6.45.
When one is subtracted from three times a number, the result is eight less than six times the original number.
Exercise 11.6.46. (Go to Solution)
When a number is subtracted from six, the result is four more than the original number.
Exercise 11.6.47.
When a number is subtracted from twenty-four, the result is six less than twice the number.
Exercise 11.6.48. (Go to Solution)
A number is subtracted from nine. This result is then increased by one. The result is eight more than three times the number.
Exercise 11.6.49.
Five times a number is increased by two. This result is then decreased by three times the number. The result is three more than three times the number.
Exercise 11.6.50. (Go to Solution)
Twice a number is decreased by seven. This result is decreased by four times the number. The result is negative the original number, minus six.
Exercise 11.6.51.
Fifteen times a number is decreased by fifteen. This result is then increased by two times the number. The result is negative five times the original number minus the opposite of ten.
Exercises for Review
Solutions to Exercises
11.7. Applications II: Solving Problems*
Section Overview
The Five-Step Method
Number Problems
Geometry Problems
The Five Step Method
We are now in a position to solve some applied problems using algebraic methods. The problems we shall solve are intended as logic developers. Although they may not seem to reflect real situations, they do serve as a basis for solving more complex, real situation, applied problems. To solve problems algebraically, we will use the five-step method.
When solving mathematical word problems, you may wish to apply the following "reading strategy." Read the problem quickly to get a feel for the situation. Do not pay close attention to details. At the first reading, too much attention to details may be overwhelming and lead to confusion and discouragement. After the first, brief reading, read the problem carefully in phrases. Reading phrases introduces information more slowly and allows us to absorb and put together important information. We can look for the unknown quantity by reading one phrase at a time.
Five-Step Method for Solving Word Problems
Let
(or some other letter) represent the unknown quantity.Translate the words to mathematical symbols and form an equation. Draw a picture if possible.
Solve the equation.
Check the solution by substituting the result into the original statement, not equation, of the problem.
Write a conclusion.
If it has been your experience that word problems are difficult, then follow the five-step method carefully. Most people have trouble with word problems for two reasons:
They are not able to translate the words to mathematical symbols. (See Section 11.5.)
They neglect step 1. After working through the problem phrase by phrase, to become familiar with the situation,
INTRODUCE A VARIABLE
Number Problems
Sample Set A
Example 11.44.
What number decreased by six is five?
Let
represent the unknown number.- Translate the words to mathematical symbols and construct an equation. Read phrases.
- Solve this equation.
Add 6 to both sides.
- Check the result.
When 11 is decreased by 6, the result is
, which is equal to 5. The solution checks. The number is 11.
Example 11.45.
When three times a number is increased by four, the result is eight more than five times the number.
Let
the unknown number.- Translate the phrases to mathematical symbols and construct an equation.
Check this result. Three times
is 
. Increasing 
by 4 results in
. Now, five times 
is 
.
Increasing 
by
results in
. The results agree, and the solution checks.
The number is
Example 11.46.
Consecutive integers have the property that if
Consecutive odd or even integers have the property that if
The sum of three consecutive odd integers is equal to one less than twice the first odd integer. Find the three integers.
- Translate the words to mathematical symbols and construct an equation. Read phrases.
- Check this result.
The sum of the three integers is
One less than twice the first integer is
. Since these two results are equal, the solution checks. The three odd integers are -7, -5, -3.
Practice Set A
Exercise 11.7.1. (Go to Solution)
When three times a number is decreased by 5, the result is -23. Find the number.
Let
Check:
The number is __________.
Exercise 11.7.2. (Go to Solution)
When five times a number is increased by 7, the result is five less than seven times the number. Find the number.
Let
Check:
The number is __________.
Exercise 11.7.3. (Go to Solution)
Two consecutive numbers add to 35. Find the numbers.
Check:
The numbers are __________ and __________.
Exercise 11.7.4. (Go to Solution)
The sum of three consecutive even integers is six more than four times the middle integer. Find the integers.
Let
smallest integer.
__________= next integer.
__________= largest integer.Check:
The integers are __________, __________, and __________.
Geometry Problems
Sample Set B
Example 11.47.
The perimeter (length around) of a rectangle is 20 meters. If the length is 4 meters longer than the width, find the length and width of the rectangle.
Let
the width of the rectangle. Then,
the length of the rectangle.-
We can draw a picture.
The length around the rectangle is
- Check:
The length of the rectangle is 7 meters. The width of the rectangle is 3 meters.
Practice Set B
Exercise 11.7.5. (Go to Solution)
The perimeter of a triangle is 16 inches. The second leg is 2 inches longer than the first leg, and the third leg is 5 inches longer than the first leg. Find the length of each leg.
Let
length of the first leg.
__________= length of the second leg.
__________= length of the third leg.We can draw a picture.
Check:
The lengths of the legs are__________,__________, and__________.
Exercises
For the following 17 problems, find each solution using the five-step method.
Exercise 11.7.6. (Go to Solution)
What number decreased by nine is fifteen?
Let
the number. Check:
The number is__________.
Exercise 11.7.7.
What number increased by twelve is twenty?
Let
the number. Check:
The number is__________.
Exercise 11.7.8. (Go to Solution)
If five more than three times a number is thirty-two, what is the number?
Let
the number. Check:
The number is__________.
Exercise 11.7.9.
If four times a number is increased by fifteen, the result is five. What is the number?
Let
Check:
The number is__________.
Exercise 11.7.10. (Go to Solution)
When three times a quantity is decreased by five times the quantity, the result is negative twenty. What is the quantity?
Let
Check:
The quantity is__________.
Exercise 11.7.11.
If four times a quantity is decreased by nine times the quantity, the result is ten. What is the quantity?
Let
Check:
The quantity is__________.
Exercise 11.7.12. (Go to Solution)
When five is added to three times some number, the result is equal to five times the number decreased by seven. What is the number?
Let
Check:
The number is__________.
Exercise 11.7.13.
When six times a quantity is decreased by two, the result is six more than seven times the quantity. What is the quantity?
Let
Check:
The quantity is__________.
Exercise 11.7.14. (Go to Solution)
When four is decreased by three times some number, the result is equal to one less than twice the number. What is the number?
Check:
Exercise 11.7.15.
When twice a number is subtracted from one, the result is equal to twenty-one more than the number. What is the number?
Exercise 11.7.16. (Go to Solution)
The perimeter of a rectangle is 36 inches. If the length of the rectangle is 6 inches more than the width, find the length and width of the rectangle.
Let
the width.
__________= the length.
We can draw a picture.
Check:
The length of the rectangle is__________ inches, and the width is__________inches.
Exercise 11.7.17.
The perimeter of a rectangle is 48 feet. Find the length and the width of the rectangle if the length is 8 feet more than the width.
Let
the width.
__________= the length.We can draw a picture.
Check:
The length of the rectangle is__________feet, and the width is__________feet.
Exercise 11.7.18. (Go to Solution)
The sum of three consecutive integers is 48. What are they?
Let
the smallest integer.
__________= the next integer.
__________= the next integer. Check:
The three integers are __________,__________, and __________.
Exercise 11.7.19.
The sum of three consecutive integers is -27. What are they?
Let
the smallest integer.
__________= the next integer.
__________= the next integer. Check:
The three integers are __________, __________, and __________.
Exercise 11.7.20. (Go to Solution)
The sum of five consecutive integers is zero. What are they?
Let
The five integers are __________, __________, __________, __________, and __________.
Exercise 11.7.21.
The sum of five consecutive integers is -5. What are they?
Let
The five integers are __________, __________,__________,__________, and __________.
Continue using the five-step procedure to find the solutions.
Exercise 11.7.22. (Go to Solution)
The perimeter of a rectangle is 18 meters. Find the length and width of the rectangle if the length is 1 meter more than three times the width.
Exercise 11.7.23.
The perimeter of a rectangle is 80 centimeters. Find the length and width of the rectangle if the length is 2 meters less than five times the width.
Exercise 11.7.24. (Go to Solution)
Find the length and width of a rectangle with perimeter 74 inches, if the width of the rectangle is 8 inches less than twice the length.
Exercise 11.7.25.
Find the length and width of a rectangle with perimeter 18 feet, if the width of the rectangle is 7 feet less than three times the length.
Exercise 11.7.26. (Go to Solution)
A person makes a mistake when copying information regarding a particular rectangle. The copied information is as follows: The length of a rectangle is 5 inches less than two times the width. The perimeter of the rectangle is 2 inches. What is the mistake?
Exercise 11.7.27.
A person makes a mistake when copying information regarding a particular triangle. The copied information is as follows: Two sides of a triangle are the same length. The third side is 10 feet less than three times the length of one of the other sides. The perimeter of the triangle is 5 feet. What is the mistake?
Exercise 11.7.28. (Go to Solution)
The perimeter of a triangle is 75 meters. If each of two legs is exactly twice the length of the shortest leg, how long is the shortest leg?
Exercise 11.7.29.
If five is subtracted from four times some number the result is negative twenty-nine. What is the number?
Exercise 11.7.30. (Go to Solution)
If two is subtracted from ten times some number, the result is negative two. What is the number?
Exercise 11.7.31.
If three less than six times a number is equal to five times the number minus three, what is the number?
Exercise 11.7.32. (Go to Solution)
If one is added to negative four times a number the result is equal to eight less than five times the number. What is the number?
Exercise 11.7.33.
Find three consecutive integers that add to -57.
Exercise 11.7.35.
Find three consecutive even integers that add to -24.
Exercise 11.7.37.
Suppose someone wants to find three consecutive odd integers that add to 120. Why will that person not be able to do it?
Exercise 11.7.38. (Go to Solution)
Suppose someone wants to find two consecutive even integers that add to 139. Why will that person not be able to do it?
Exercise 11.7.39.
Three numbers add to 35. The second number is five less than twice the smallest. The third number is exactly twice the smallest. Find the numbers.
Exercise 11.7.40. (Go to Solution)
Three numbers add to 37. The second number is one less than eight times the smallest. The third number is two less than eleven times the smallest. Find the numbers.
Exercises for Review
Exercise 11.7.42. (Go to Solution)
(Section 7.4) A 5-foot woman casts a 9-foot shadow at a particular time of the day. How tall is a person that casts a 10.8-foot shadow at the same time of the day?
Exercise 11.7.45.
(Section 11.6) Twice a number is added to 5. The result is 2 less than three times the number. What is the number?
Solutions to Exercises
Solution to Exercise 11.7.26. (Return to Exercise)
The perimeter is 20 inches. Other answers are possible. For example, perimeters such as 26, 32 are possible.
Solution to Exercise 11.7.38. (Return to Exercise)
…because the sum of any even number (in this case, 2) o even integers (consecutive or not) is even and, therefore, cannot be odd (in this case, 139)
11.8. Summary of Key Concepts*
Summary of Key Concepts
A numerical expression results when numbers are associated by arithmetic operation signs. The expressions
,
,
and
are numerical expressions.
When an arithmetic operation sign connects a letter with a number or a letter with a letter, an algebraic expression results. The expressions
,
,
, and
are algebraic expressions.
Terms are parts of sums and are therefore separated by addition (or subtraction) signs. In the expression,
,
and
are the terms. Factors are parts of products and are therefore separated by multiplication signs. In the expression 
, 5 and 
are the factors.
The coefficient of a quantity records how many of that quantity there are. In the expression
, the coefficient 7 indicates that there are seven
's.
Numerical evaluation is the process of determining the value of an algebraic expression by replacing the variables in the expression with specified values.
An algebraic expression may be simplified by combining like terms. To combine like terms, we simply add or subtract their coefficients then affix the variable. For example
.
An equation is a statement that two expressions are equal. The statements
and
are equations. The expressions represent the same quantities.
A conditional equation is an equation whose truth depends on the value selected for the variable. The equation
is a conditional equation since it is only true on the condition that 3 is selected for
.
The values that when substituted for the variables make the equation true are called the solutions of the equation. An equation has been solved when all its solutions have been found.
Equations that have precisely the same solutions are called equivalent equations. The equations
and
are equivalent equations.
Given any equation, we can obtain an equivalent equation by
adding the same number to both sides, or
subtracting the same number from both sides.
To solve
, subtract
from both sides.
To solve
,
add
to both sides.
Given any equation, we can obtain an equivalent equation by
multiplying both sides by the same nonzero number, that is, if
,
and
are equivalent.dividing both sides by the same nonzero number, that is, if
,
and
are equivalent.
To solve
,
, divide both sides by
.
To solve
,
,
multiply both sides by
.
In solving applied problems, it is important to be able to translate phrases and sentences to mathematical expressions and equations.
To solve problems algebraically, it is a good idea to use the following five-step procedure. After working your way through the problem carefully, phrase by phrase:
Let
(or some other letter) represent the unknown quantity.Translate the phrases and sentences to mathematical symbols and form an equation. Draw a picture if possible.
Solve this equation.
Check the solution by substituting the result into the original statement of the problem.
Write a conclusion.
11.9. Exercise Supplement *
Exercise Supplement
Algebraic Expressions (Section 11.2)
For problems 1-10, specify each term.
Exercise 11.9.2.
Exercise 11.9.4.
Exercise 11.9.6.
Exercise 11.9.8.
Exercise 11.9.10.
Exercise 11.9.12.
Write
in a simpler way.
Exercise 11.9.14.
In the expression
, how many
’s are indicated?
Exercise 11.9.16.
In the expression
, how many
’s are indicated?
For problems 17-46, find the value of each expression.
Exercise 11.9.18.
, if
and
Exercise 11.9.20.
, if
,
, and
Exercise 11.9.22.
, if
and
Exercise 11.9.24.
, if
Exercise 11.9.26.
, if
and
Exercise 11.9.28.
, if
and
Exercise 11.9.30.
, if
Exercise 11.9.32.
, if
Exercise 11.9.34.
, if
Exercise 11.9.36.
, if
Exercise 11.9.38.
, if
Exercise 11.9.40.
, if
Exercise 11.9.42.
, if
and
Exercise 11.9.44.
, if
and
Exercise 11.9.46.
, if
Combining Like Terms Using Addition and Subtraction (Section 11.3)
For problems 47-56, simplify each expression by combining like terms.
Exercise 11.9.48.
Exercise 11.9.50.
Exercise 11.9.52.
Exercise 11.9.54.
Exercise 11.9.56.
Equations of the Form 
and 
, Translating Words to Mathematical Symbols , and Solving Problems (Section 11.5,Section 11.6,Section 11.7)
For problems 57-140, solve each equation.
Exercise 11.9.58.
Exercise 11.9.60.
Exercise 11.9.62.
Exercise 11.9.64.
Exercise 11.9.66.
Exercise 11.9.68.
Exercise 11.9.70.
Exercise 11.9.72.
Exercise 11.9.74.
Exercise 11.9.76.
Exercise 11.9.78.
Exercise 11.9.80.
Exercise 11.9.82.
Exercise 11.9.84.
Exercise 11.9.86.
Exercise 11.9.88.
Exercise 11.9.90.
Exercise 11.9.92.
Exercise 11.9.94.
Exercise 11.9.96.
Exercise 11.9.98.
Exercise 11.9.100.
Exercise 11.9.102.
Exercise 11.9.104.
Exercise 11.9.106.
Exercise 11.9.108.
Exercise 11.9.110.
Exercise 11.9.112.
Exercise 11.9.114.
Exercise 11.9.116.
Exercise 11.9.118.
Exercise 11.9.120.
Exercise 11.9.122.
Exercise 11.9.124.
Exercise 11.9.126.
Exercise 11.9.128.
Exercise 11.9.130.
Exercise 11.9.132.
Exercise 11.9.134.
Exercise 11.9.136.
Exercise 11.9.138.
Exercise 11.9.140.
Solutions to Exercises
11.10. Proficiency Exam*
Proficiency Exam
For problems 1 and 2 specify each term.
For problems 4-9, find the value of each expression.
For problems 10-12, simplify each expression by combining like terms.
For problems 13-22, solve each equation.
Exercise 11.10.23. (Go to Solution)
(Section 11.6 and Section 11.7) Three consecutive even integers add to -36. What are they?
Exercise 11.10.24. (Go to Solution)
(Section 11.6 and Section 11.7) The perimeter of a rectangle is 38 feet. Find the length and width of the rectangle if the length is 5 feet less than three times the width.
Exercise 11.10.25. (Go to Solution)
(Section 11.6 and Section 11.7) Four numbers add to -2. The second number is three more than twice the negative of the first number. The third number is six less than the first number. The fourth number is eleven less than twice the first number. Find the numbers.
