Example: 2x-1=y,2y+3=x
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Evaluating Algebraic Expressions, Combining Like Terms, Writing Algebraic Expressions, Solving equations, Formulas and Problems
3.1 Evaluating Algebraic Expressions
The same set of rules for order of operations for whole numbers discussed in
Section 1.2 is also used with integers. The rules are restated here for easy
reference.
Rules for Order of Operations
1. Work within symbols of inclusion (parentheses, brackets, or braces),beginning with the innermost pair.
2. Evaluate the terms with exponents.
3. From left to right, perform multiplications and divisions as they appear.
4. From left to right, perform additions and subtractions as they appear.
Examples
1. 8÷4-3^2
8÷4-3^2=8÷4-9 exponents
=2-9 division
=-7 subtraction
2. (7+8)÷5*4-20
(7+8)÷5*4-20=15÷5*4-20 parentheses
=3*4-20 division
=12-20 multiplication
=-8 subtraction
3. 4*5-(6*2-3)+4÷2
4*5-(6*2-3)+4÷2=4*5-(9)+4÷2 parentheses
=20-9+2 multiplication and division
=13 subtraction and addition
4. (-5-6)/11+3(-5)
(-5-6)/11+3(-5)=-11/11+3(-5) fraction bar is a symbol of inclusion
=-1+(-15) division and multiplication
=-16 addition
Practice Quiz
| Questions | Answers |
| Find the value of each expression | |
| 1. 7*5+3*8 | 1. 59 |
| 2. (-3)^2+4(-3)-1 | 2. -4 |
| 3. 5+(2-3)^2+14÷7 | 3. 0 |
| 4. -5(7+3)÷2*2 | 4. -50 |
| 5. (2^5+4)/6-4(-2)(The fraction bar should be treated as a symbol of inclusion.) |
5. 14 |
One situation deserves special mention. Is there a difference between (-7)^2 and -7^2, or are they the same? In the expression (-7)^2, the base is -7 and -7 is to be squared:
(-7)^2=(-7)(-7)=+49
But, for -7^2, the base is 7 and the rules for order of operations say to square 7 first:
-7^2=-(7^2)=-49
We can also think that -7^2=-1.7^2=-1*49=-49.
Suppose x=3 and we want to evaluate the expression -x^2. Then,
-x^2=-3^2=-9 or -x^2=-1*x^2=-1*3^2=-1*9=-9
If x=-3, then
-x^2=-[(-3)^2]=-[9]=-9
An expression that involves only multiplications and/or divisions with constants and/or variables is called a term. A single constant or variable is also a term. Examples of terms are:
3x,-7y,14x^2,-x/y^2,16
To evaluate an algebraic expression that involves the sums and/ or differences of several terms, substitute the chosen value for each variable throughout the expression, then apply the rules for order of operations.
Examples
Evaluate the following expressions if x=3 and y=-1
1. 2x+3=2*3+3=6+3=9
2. y^2-2y+1=(-1)^2-2(-1)+1=1+2+1=4
3. xy-y^2=3(-1)-(-1)^2=-3-1=-4
4. x^2-y=3^2-(-1)=9+1=10
5. -x^2-y^2=-3^2-(-1)^2=-9-(1)=-9-1=-10
6. 3x^2+2xy-12=3(3)^2+2(3)(-1)-12=3*9-6-12=27-6-12=9
3.2 Combining Like Terms
The expressions 3+x,5y-7x,3x+4x, and 5x^2-3x^2+2x are the sums of terms. Like terms (or similar terms) are those terms that are constants or contain variables that are of the same power in each term. Thus, 7 and -10 are like terms. Also, -2x and 5x are like terms, but -2x and 3x^2 are not like terms because x and x^2 are not the same power of x.
The numerical part of a term is called the coefficient of the variable or variables in the term. Thus, in the term 8x,8 is the coefficient of x.
Expressions with like terms can be simplified by applying the distributive property discussed in Section 1.1 to integers. The distributive property states that
a(b+c)=ab+ac
or ab+ac=a(b+c)
or ba+ca=(b+c)a
This last form is particularly useful when b and c are numerical coefficients. For example,
3x+5x=(3+5)x The coefficients are added
=8x
We say that 3x and 5x have been combined or that we have combined like terms. Like terms can be combined by adding (or subtracting) the coefficients.
Examples Combine like terms whenever possible
1. 3x-5x=(3-5)x=-2x
2. 6x-2x=(6-2)x=4x
3. 4(x-7)+5(x+1)=4x-28+5x+5 Use distributive property twice.
=4x+5x-28+5
=9x-23 Combine like terms.
4. 2x^2+3a-x^2-a=2x^2-x^2+3a-a
=(2-1)x^2+(3-1)a -x^2=-1*x^2 and -a=-1*a
=x^2+2a
5. (x+3x)/2+x=(4x)/2+x A fraction bar is a symbol of inclusion like parentheses.
=2x+x=3x
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| Practice Quiz | Questions | Answers |
| Simplify the following expressions by combining like terms. | ||
| 1. -2x-5x | 1. -7x | |
| 2. 12y+6-y+10 | 2. 11y+16 | |
| 3. 5(x-1)+4x | 3. 9x-5 | |
| 4. 2b^2-a+b^2+a | 4. 3b^2 |
3.3 Writing Algebraic Expressions
Algebra is a language of mathematicians, and in order to understand mathematics, you must understand the language. We want to be able to change English phrases into their “algebraic” equivalents, and vice versa. So, if a problem is stated in English, we can translate the phrases into algebraic symbols and proceed to solve the problem according to the rules developed for algebra.
The following examples illustrate how certain key words can be translated into algebraic symbols.
Examples
| English Phrase | Algebraic Expression |
|
1. 3 multiplied by the number represented by x The product of 3 and x 3 times x |
3x |
|
2. a number added to 3 the sum of z and 3 z plus 3 |
z+3 |
|
3. two times the quantity found by adding a number to 1 |
2(x+1) |
|
4. twice x plus 1 |
2x+1 |
|
5. the product of two numbers x times y |
xy |
|
6. the difference between 5 times a number and 2 times the same number |
5x-2x |
Certain words, such as those in boldface type in the previous examples, are the keys to the operations. Learn to look for these words and those from the following list.
| Addition | Subtraction | Multiplication | Division |
| add | subtract | multiply | divide |
| Sum | difference | product | quotient |
| plus | minus | times | |
| more than | less than | twice | |
| increased by | decreased by |
| Practice Quiz | Questions | Answers |
|
Change the following phrases to algebraic expressions |
||
| 1. 7less than a number | 1. x-7 | |
| 2. twice the product of two unknown numbers | 2. 2ab | |
| 3. the quotient of yy and 5 | 3. y/5 | |
| 4. an unknown amount less than 10 | 4. 10-x | |
| 5. 14 more than 3 times a number | 5. 3y+14 | |
| 6. the product of 5 with the difference of 2 and x | 6. 5(2-x) | |
| 7. four less than the product of 2 with x minus 3 | 7. 2(x-3)-4 | |
| 8. the sum of the product of 5 with a number and the product of 3 with that number | 8. 5x+3x |
Special mention should be made of the words “quotient” and “difference.” As illustrated in Problems 3 and 6 in the Practice Quiz, the division and subtraction are done with the values in the order they are given in the problem. For example, the difference between 3 and 5 is 3-5=-2, while the difference between 5 and 3 is 5-3=2.
3.4 Solving Equation
If an equation contains a variable, we want to find the value (or values) for the variable that will give a true equation when substituted for the variable. This value (or values) is called the solution to the equation and we have solved the equation.
Suppose we are given the equation
2x-1=x+3
If we substitute x=4, then
2*4-1=7 and 4+3=7
so 2*4-1=4+3
and x=4 is a solution
If we substitute x=5, then
2*5-1=9 and 5+3=8
but 9!=8,
so 2*5-1!=5+3
and x=5 is not a solution.
Two equations are equivalent if they have exactly the same solutions. For example, 2x-1 = x+3 and x+1=5 are equivalent since x=4 is the solution for each equation.
We need some procedures that will allow us in a step-by-step manner to find the solutions to equations that contain variables. The following two ideas are basic.
1. Whatever is done to one side of the equation must be done to the other side. (This does not include simplifying expressions and combining like terms.)
2. The object is to find a simple equation, such as x=4, that is equivalent to the original equation.
In the following examples, each equation is solved in a step-by-step manner with an explanation for each step. Study each example carefully. Note that the equivalent equations are written one under the other. Do not write several equations on one line or set one equation equal to another equation.
Examples
1. x+7=12 Write the equation.
x+7-7=12-7 Add -7 to both sides
x=5 Simplify.
2. 2x-3=13 Write the equation.
2x-3+3=13+3 Add 3 to both sides.
2x=16 Simplify.
(2x)/2=16/2 Divide both sides by 2, the coefficient of x.
x=8 Simplify.
3. 5x-1=-11 Write the equation.
5x-1+1=-11+1 Add +1 to both sides.
5x=-10 Simplify; now one side has all terms with variables and only terms with variables.
(5x)/5=-10/5 Divide both sides by 5, the coefficient of x.
x=-2 Simplify
4. 4x+1-x=13+x Write the equation.
3x+1=13+x Simplify
3x+1-1=13+x-1 Add -1 to both sides.
3x=12+x Simplify
3x-x=12+x-x Add -x to both sides.
2x=12 Simplify; now one side has all terms with variables and only terms with variables.
(2x)/2=12/2 Divide both sides by 2, the coefficient of x.
x=6 Simplify
5. (2x)/5+2=6 Write the equation.
(2x)/5+2-2=6-2 Add -2 to both sides.
(2x)/5=4 Simplify
5*(2x)/5=4*5 Multiply both sides by 5.
2x=20 Simplify
(2x)/2=20/2 Divide both sides by 2
x=10 Simplify
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Each solution may be checked by substituting it into the original equation. A true statement will result if there are no errors.
If an expression which represents a nonzero number is (a) added to, (b) divided into, or (c) multiplied by both sides of an equation, the new equation will be equivalent to the original equation.
3.5 Formulas
Formulas are general rules or principles stated mathematically. For example, in business, simple interest is the product of the principal, the rate of interest, and the time in years. The same rule mathematically is the formula {Iota}=PRT. Other formulas are given below.
A=PIr^2 The area (A) of a circle equals the product of PI and the square of the radius (r).
p=4s The perimeter (p) of a square is 4 times the length of one side (s).
A=1/2bh The area (A) of a triangle equals one-half the product of the base (b) and the height (h).
d=rt The distance traveled equals the product of the rate (r) and the time (t).
C=5/9(F-32) Temperature in centigrade (C) equals 5/9 the difference between the Fahrenheit temperature (F) and 32.
In the last formula, suppose that F = 212°, the boiling temperature of water at sea level. What would be the reading in centigrade degrees? Substituting 212° for F gives
C=5/9(212-32)=5/9(180)=100
Suppose that the question is reversed. If C = 20°, what would be the corresponding value of F? Substituting 20 for C and solving for F gives
20=5/9(F-32)
20*9/5=9/5*5/9(F-32) Multiply both sides by 9/5, the reciprocal 5/9.
36=F-32
68=F
Solving for F in terms of C can be done in the following manner:
C=5/9(F-32)
9/5*C=9/5*5/9(F-32)
9/5C=F-32
9/5C+32=F
The formula solved for C is
C=5/9(F-32)
and solved for F is
F=9/5C+32
The object here is to solve formulas for one of the variables in terms of the other variables. That is, using the techniques of solving equations which have only one variable, treat the other variables as constants and solve for the desired variable.
Examples
1. Given P=a+b+c Formula for the perimeter of a triangle
Solve for b.
Solution:
P=a+b+c
P-a-c=a+b+c-a-c
P-a-c=b
2. Given P=2l+2w Formula for the perimeter of a rectangle
Solve for l
Solution:
P=2l+2w
P-2w=2l+2w-2w
P-2w=2l
(P-w)/2=(2l)/2
(P-2w)/2=l
or
P=2l+w
P/2=(2l+2w)/2
P/2=(2l)/2+(2w)/2
P/2=l+w
P/2-w=l
Both answers are correct since P/2-w=P/2-(2w)/2=(P-2w)/2
3. Given 3y-4x+9=0, solve for y.
Solution:
3y-4x+9=0
3y-4x+9+4x-9=0+4x-9
3y=4x-9
y=(4x-9)/3 or y=(4x)/3-9/3=4/3x-3
Either of these forms is correct. If you have one answer and the other is in the Answer Key, you should recognize your answer as being correct but in a different form.
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4. Given V=T/(P-5), solve P.
Solution:
V=T/(P-5)
V(P-5)=T/(P-5)(P-5) Multiply both sides by P-5.
V(P-5)=T
(V(P-5))/V=T/V
P-5=T/V
P=T/V+5
3.6 Word Problems (Numbers and Consecutive Integers)
In Section 3.3 we discussed translating English phrases into algebraic expressions‘ The phrase “8 added to twice a number" translates algebraically to 2x + 8. How do you translate “4 more than a number?” If you said x + 4, you are correct. Now, the object is to translate an entire sentence into an equation and then to solve the equation. The two phrases above might be involved in a sentence such as the following:
“If 8 is added to twice a number, the result is 4 more than the number.”
Algebraically,
2x+8=x+4 "the result is"translates as=
Solving,
2x+8=x+4
2x+8-x=x+4-x
x+8=4
x+8-8=4-8
x=-4
In this section, the word problems will be simply exercises in translating sentences into equations and solving these equations. More sophisticated “application” problems will be discussed in later chapters. Such problems will involve geometric formulas, distance, interest, work, inequalities, and mixture.
Examples
1. Three times the sum of a number and 5 is equal to twice the number plus 5. Find the number.
Solution Let x : the unknown number.
| 3 times the sum of a number and 5 | is equal to | twice the number plus 5 |
| 3(x+5) | = | 2x+5 |
| 3x+15 | = | 2x+5 |
| 3x+15-2x | = | 2x+5-2x |
| x+5 | = | 5 |
| x+15-15 | = | 5-15 |
| x | = | -10 |
The number is -10.
2. If a number is decreased by 36 and the result is 76 less than twice the number, what is the number?
Solution Let n = the unknown number.
| a number decreased by 36 | the result is | 76 less than twice the number |
| n-36 | = | 2n-76 |
| n-36-n | = | 2n-76-n |
| -36 | = | n-76 |
| -36+76 | = | n-76+76 |
| 40 | = | n |
The number is 40.
Consecutive integers are two integers that differ by 1, or the second integer is 1 more than the first integer. For example, 21 and 22 are consecutive integers. -14 and -13 are consecutive integers. In general, if n is one integer, then n + 1 is the next consecutive integer.
An example of three consecutive integers is 51,52,53. Another example is -9,-8,-7. If n is one integer, then n + 1 is the next consecutive integer and n+2 is the third consecutive integer.
Consecutive even integers are even integers that differ by 2; that is, the second integer is 2 more than the first. For example, 36 and 38 are two consecutive even integers. Also, -12,-10 and -8 are three consecutive even integers If n is an even integer, then n+2 is the next consecutive even integer and n+4 is the third consecutive even integer.
Consecutive odd integers are odd integers that differ by 2; again, the second integer is 2 more than the first. For example, -15 and -13 are two consecutive odd integers. Also, 17,19, and 21 are three consecutive odd integers. If n is an odd integer, then n+2 is the next consecutive odd integer and n + 4 is the third consecutive odd integer.
Examples: Consecutive Integers
1. Find three consecutive integers such that the sum of the first and third is 76less than three times the second.
Let n = the first integer
n + 1 = the second integer
n + 2 = the third integer
n+(n+2)=3(n+1)-76
2n+2=3n+3-76
2n+2=3n-73
2n+2+73-2n=3n-73+73-2n
75=n
76=n+1
77=n+2
The three consecutive integers are 75,76, and 77.
2. Three consecutive odd integers are such that their sum is -3. What are the integers?
Let n = the first odd integer
n + 2 = the second odd integer
n + 4 = the third odd integer
n+(n+2)+(n+4)=-3
3n+6=-3
3n=-9
n=-3
n+2=-1
n+4=+1
The three consecutive odd integers are -3, -1, and +1.