Algebra is a branch of mathematics that deals with unknown quantities. Mastery of mathematics depends on a sound understanding of algebra. Engineers and scientists use algebra every day. Business and industry rely on algebra to help solve many problems. Because of its importance in modern living, algebra is studied in schools and colleges in all parts of the world.
Unknown numbers in algebra are represented by letters, such as x or y. In some problems, the letter can be replaced by only one number. A simple example would be x + 3 = 8. For this statement to be true, x must be 5, because 5 + 3 = 8. In other problems, the letter may be replaced by one of many numbers. For example, for the algebraic statement x + y = 12 to be true, x would be 6 if y is 6, or x would be 4 if y is 8. In such an algebraic statement, you can find many values for x that make true statements if you give different values to y.
People can use algebra to solve problems in ways that are beyond the range of arithmetic alone. For example:
An airplane travels 1,710 kilometers in 4 hours flying with the wind, but it travels only 1,370 kilometers in 5 hours flying against the wind. The speed of the airplane in relation to the air is the same in both directions, and the wind speed is constant. What is (1) the speed of the airplane in relation to the air and (2) the speed of the wind?
The key to solving this problem is to use letters to represent the two unknown numbers. For example, you might use x to represent the speed of the plane relative to the air, and y to represent wind speed. Using letters in this way is not part of arithmetic, but is an essential part of algebra.
Learning algebra
Sets and variables.
Letters in algebra are related to sets of numbers. Everyone is familiar with sets of objects. There are sets of books, sets of stamps, and sets of dishes. Sets of numbers are much the same. In algebra, one way to describe a set of numbers is to use a capital letter, such as N, as a name for the set. Then you list the numbers of the set within braces { }. For example, here is the set of single-digit whole numbers:
W = {0,1,2,3,4,5,6,7,8,9}
Here is the set of odd numbers smaller than 20:
Y = {1,3,5,7,9,11,13,15,17,19}
These are the kinds of sets used in algebra.
Imagine a group of people whose ages are 12 years, 15 years, 20 years, and 24 years. You can write these ages as a set of numbers:
A = {12,15,20,24}
How old will these people be three years from now? One way of answering this question is to write out 12 + 3, 15 + 3, 20 + 3, and 24 + 3. However, the number 3 is the same in all four of these expressions. In algebra, you can write all four expressions as one general expression, m + 3, in which m can be replaced with any number of the set A. For example, you can replace m with 12, 15, 20, or 24.
The letter m is called a variable, and the set A is the domain of the variable. The number 3 in the expression m + 3 is called a constant, because 3 always has the same value. A variable in algebra is a letter that can be replaced by one or more numbers belonging to a set.
Statements and equations.
In mathematics, a statement is a sentence that is either true or false. Mathematical statements can be illustrated in everyday language. For example, here is an incomplete statement:
“_____ was the inventor of the telephone.”
As it stands, this statement is neither true nor false. Suppose you write a name in the blank:
“Bell was the inventor of the telephone.”
Now the statement is true.
You can write a statement with a variable:
“y is a state bordered by the Pacific Ocean.”
You can replace a variable with the members of its domain. That is, you can replace the variable with names that will produce a true or false statement.
“Ohio is a state bordered by the Pacific Ocean.”
This statement is false. It is true only when you use Alaska, California, Hawaii, Oregon, or Washington:
“Oregon is a state bordered by the Pacific Ocean.”
The replacements that make true statements are called roots. The set that includes all the roots is called the solution set. As with other sets, braces are used to enclose the solution set. The solution set of this example is {Alaska, California, Hawaii, Oregon, Washington}. In algebra, you do not use names to replace a variable. Instead, you use numbers.
Equations are one kind of sentence in algebra. They are mathematical sentences that say two things are equal. Here is a simple equation:
7 + x = 12
This equation means that “the sum of 7 and a number equals 12.” To solve the equation, you can replace x with different numbers until you find one that will make the equation a true statement. If you replace x in this equation with 5, the equation will be a true statement. If you replace x with any other number, the equation will be false. So the solution set to this equation is {5}. The solution set consists of only one root.
Equations can have more than one root:
x2 + 18 = 9x
The little 2 above the first x means that the number x represents is squared. That is, the number is multiplied once by itself (see Square ). Also, one quantity placed next to another quantity indicates that one quantity is to be multiplied by the other quantity. Therefore, the expression9x_means _x multiplied by 9.
In the above equation, you can replace x with 3:
You can also replace x with 6:
Any other replacements of x make the equation a false statement. So 3 and 6 are roots of the equation, and its solution set is {3,6}.
Some equations do not have roots:
x = x + 3
This equation becomes a false statement for any number you use to replace x. Its solution set is called an empty set. An empty set is written { }.
Some equations have many roots. Some even have an infinite (unlimited) number of roots:
(x + 1)2 = x2 + 2x + 1
This equation will be a true statement if you replace x with any number. Its solution set consists of all numbers.
Mathematicians use a number of terms to describe parts of an equation. They call the expression on each side of the equals sign a member of the equation. For example, in the equation 3x + 2 = 11, 3x + 2 is the left member and 11 is the right member. Each part of a member that is connected by addition or subtraction signs—or stands alone—is called a term. Therefore, 3x and 2 are the terms in the left member, and 11 is the term in the right member.
Solving equations.
The goal in solving an equation with a variable is to isolate the variable on one side of the equation. It does not matter on which side of the equals sign the variable appears because x = 5 means 5 = x. But most people prefer to have the variable on the left because they read from left to right.
A variable may be isolated by means of subtraction, division, addition, and multiplication. Sometimes, you must perform more than one operation to arrive at the final answer.
Subtraction.
If the same number is subtracted from each side of an equation, the new members remain equal. All roots of the original equation are also roots of the new equation. Thus, for example, you can subtract 2 from each member of the equation 3x + 2 = 11:
3x + 2 – 2 = 11 – 2
to obtain the new equation
3x = 9
The equation 3x = 9 is equivalent to 3x + 2 = 11. The roots of either of these equations will solve the other. To isolate the variable of the new equation, you must perform one more operation—the operation of division.
Division.
If each side of an equation is divided by the same number, except zero, the new members will be equal. The roots of the original equation are the roots of the new equation. Using this rule you can divide each side of 3x = 9 by 3:
So the solution set of the equation 3x + 2 = 11 is {3}. You can prove this by replacing x with 3 in the original equation: 3 X 3 + 2 = 11, or 11 = 11.
You cannot divide members of an equation by zero. Division by zero is meaningless.
Addition.
Another rule for solving algebraic equations is that if the same number is added to each member of an equation, the new members will be equal. The roots of the original equation are roots of the new equation. For example, in the equation x – 6 = 18, you can add 6 to each member of the equation to isolate the x on the left side of the equation. That is, x – 6 + 6 = 18 + 6, and x = 24. The solution is the set {24}.
When adding terms with identical variables, the numbers before the variables are added. For example, 5a + 2a = 7a. When subtracting terms with identical variables, the numbers are subtracted, so that 8y – 3y = 5y.
Multiplication.
The last rule for solving simple equations is that if each member of an equation is multiplied by the same number, the new members are equal. It would not be useful to multiply by zero, however, because any number multiplied by zero equals zero.
After multiplying both sides of an equation by the same number, the roots of the original equation equal the roots of the new equation. For example, you can multiply each member of the equation 1/4x = 5 by 4:
4 X 1/4x = 4 X 5
to obtain
x = 20
Thus, the solution set of 1/4x = 5 is {20}.
You can use all four rules to find the solution set of the equation
2/3x – 4 = 1/4x + 6
First, use multiplication to produce an equation that has only whole numbers. Such an equation is easier to solve than one containing fractions. The numbers in the denominators, 3 and 4, have the common factor 12 (see Factor ). Multiplying both sides of the equation by 12 therefore changes the fractions into whole numbers:
12(2/3x – 4) = 12(1/4x + 6) 8x – 48 = 3x + 72
Second, add 48 to each member of the equation to eliminate the 48 from the left side of the equation:
8x – 48 + 48 = 3x + 72 + 48 8x = 3x + 120
Third, subtract 3x from each member to eliminate the 3x from the right side of the equation:
8x – 3x = 3x + 120 – 3x 5x = 120
Finally, divide each member by 5 to isolate the variable x on the left side of the equation:
The solution set is therefore {24}. You can verify this by replacing x with 24 in the original equation:
2/3 X 24 – 4 = 1/4 X 24 + 6 16 – 4 = 6 + 6 12 = 12
Since the equation-solving techniques did not produce any other solutions, 24 is the only solution.
Positive and negative numbers.
In arithmetic, you can always add, multiply, or divide numbers. But you cannot always subtract. For example, “3 – 5” is meaningless in ordinary arithmetic. Algebra has an extended number system that solves this problem.
In ordinary arithmetic, numbers indicate only size. That is, they show how many or how much. But many everyday measurements indicate both size and direction. The temperature above or below zero is a good example of this. In algebra, we use numbers that show direction.
You can show these new numbers on a scale:
The origin or starting point is zero. To the right of zero, the points show positive distance or direction. These numbers are like temperatures above zero. To the left of zero, the points show negative distance or direction. These numbers are like temperatures below zero. Point A is not just 1, but +1 or positive one. The + sign shows its direction from zero. Point B is not just 1, but –1 or negative one. The – sign also shows direction. The numbers on this scale are called positive numbers and negative numbers. In everyday life, you can use these numbers to represent temperatures, distances above or below sea level, changes in stock-market prices, business earnings, and many other things. For every positive number, there is a negative number of the same arithmetical size. For example, the number 7 always means seven things, positive or negative. The arithmetical size of a number is called its absolute value.
You can add, subtract, multiply, and divide positive and negative numbers, but the rules of these operations are different from those in ordinary arithmetic.
Adding
can be illustrated with the problem (+5) + (–7), or the sum of positive five and negative seven. You can work out the solution on the following scale.
If you were adding (+5) and (+7) on a scale, you would count five points to the right from zero, and then count seven more to the right to (+12). To add (+5) + (–7), start at zero on the scale above and count off the first number to be added. This number is +5, so you count off 5 to the right. Next, count off in the direction indicated by the second number to be added. This number is –7, so you count off –7 to the left from +5. This takes you to the left of 0, to –2. You can read the sum of (+5) + (–7) on the scale: –2. Therefore, (+5) + (–7) = (–2). Numbers with positive or negative signs are often called signed numbers.
One rule for adding signed numbers has two parts. First, if the signs are the same, add the absolute values of the numbers and give the sum the common sign. For example,
(+5) + (+8) = (+13) and (–5) + (–8) = (–13)
Second, if the signs are different, subtract the smaller absolute value from the larger absolute value, and give the result the sign of the number with the larger absolute value. For example,
(+5) + (–8) = (–3) and (–5) + (+8) = (+3)
Subtracting.
In arithmetic, subtraction is often thought of as the opposite of addition. But using negative numbers, we can think about subtraction as simply a special kind of addition. Subtracting a number is the same as adding the number with the opposite sign.
You may have noticed this rule when adding negative numbers to positive numbers. For example, take the arithmetic expression 9 – 4. In this expression, both the 9 and the 4 are positive. By switching the sign of the number 4 from positive to negative, we can rewrite this subtraction problem as an addition problem:
(+9) – (+4) becomes (+9) + (–4)
Both expressions equal the same number: 5. The idea of switching signs lets us simplify how we subtract negative numbers. Subtracting a negative number is the same as adding that number’s positive. For example, the expression (+6) – (–2) is the same as (+6) + (+2). Both expressions equal 8.
Multiplying.
The rule for multiplying signed numbers is to multiply the absolute values. If the signs are alike, the product is positive.
(+3) X (+8) = (+24) (–3) X (–8) = (+24)
If the signs are not alike, the product is negative.
(+3) X (–8) = (–24) (–3) X (+8) = (–24)
Dividing.
The rule for dividing signed numbers is similar. Dividing numbers with the same sign gives a positive quotient.
(+24) ÷ (+3) = (+8) (–24) ÷ (–8) = (+3)
Dividing numbers with signs that are not alike gives a negative quotient.
(+24) ÷ (–3) = (–8) (–24) ÷ (+8) = (–3)
In algebra, you may need to use negative numbers to obtain a root to an equation. For example, the equation x + 4 = 1 has no root with positive numbers. With the extended number system, its root is –3. Operations with negative numbers can be applied to variables that represent numbers. That is, you can deal with quantities such as –x or –y.
Writing formulas.
Algebra uses general formulas to help solve many practical problems in science, engineering, and everyday life. A wide variety of arithmetic situations can be expressed in general formulas.
One example of the use of general formulas involves room dimensions. Consider a room that is 5 meters long and 4 meters wide. Its perimeter, or outside measurement, is 5 + 4 + 5 + 4 meters, or 2 X (5 + 4) meters. If the room is 5 meters long and the width is unknown, you can use w, a variable, to represent the width. The perimeter is then 5 + w + 5 + w, or 2 X (5 + w). Going one step further, you can write a formula for the perimeter of any rectangular room by using l for the length and w for the width. The formula is 2 X (l + w). You can solve many problems with this kind of formula.
Some situations call for an equation. For example, a man collected a sum of money on August 1, and 1/3 as much on August 2. He collected a total of $6,500. How much was each amount? If n is the amount collected on August 1, then 1/3n is the amount collected on August 2. The equation is n + 1/3n = 6,500. You can solve this equation to find n. First, multiply both members of the equation by 3 to change the fraction 1/3 to a whole number:
3 X (n + 1/3n) = 3 X 6,500 3n + n = 19,500
Add the terms on the left side of the equation:
4n = 19,500
Divide both members of the equation by 4 to find n:
n = 4,875
And 1/3n = 4,875 ÷ 3, or 1,625. Therefore, the man collected $4,875 on August 1 and $1,625 on August 2. To check this result, add the two amounts collected: $4,875 + $1,625 = $6,500.
Basic algebra
After you learn to work with variables, equations, and signed numbers, you will find that the fundamental principles of algebra are not hard to understand.
Symbols in algebra.
The symbol + indicates addition. But in algebra, it also signifies a positive number. The symbol – indicates subtraction and a negative number. You usually do not use X to indicate multiplication in algebra, because it might be confused with the letter x. Instead, you use a dot · or no symbol at all. You write a multiplied by b as a · b, (a)(b), or ab. (Note that 3 · 6 and (3)(6) both mean six multiplied by three, but that 36 still means 36, as in arithmetic.) The symbol ÷ for division is the same as it is in arithmetic.
Parentheses (), brackets [], and braces {} often enclose quantities or numbers. They are called signs of aggregation because everything within them must be treated as a single expression. You must often simplify the enclosed expression before it can be used in other parts of a problem. Here is an example using numbers:
{12 + [4 + 5 – (5 – 3) + 4] – 4}
First, simplify the group (5 – 3):
{12 + [4 + 5 – 2 + 4] – 4}
Second, simplify the group [4 + 5 – 2 + 4]:
{12 + 11 – 4} = 19
You use the same method to simplify expressions with variables. Here is an example of simplifying groups of variables:
{5a + 6a + [5a – a + (3a + 4a)] – a}
First, simplify the group (3a + 4a):
{5a + 6a + [5a –a + 7a] – a}
Second, simplify the group [5a – a + 7a]:
{5a + 6 a + 11a – a} = 21a
Sometimes it is useful to remove the parentheses from an expression without simplifying it. You can do this by using the rules for addition and subtraction of signed numbers. For example, the expression a + (b + c) can be rewritten a + b + c. To illustrate this, the expression 40 + (8 – 2) means that 8 – 2, or 6, must be added to 40, or 40 + 6. Removing the parentheses, 40 + 8 – 2, or 48 – 2, is the same as the simplified expression, 40 + 6. If an expression within parentheses has an addition or positive sign before it, you can remove the parentheses without changing the signs of the quantities within the parentheses. Thus, a + (–b – c) becomes a – b – c.
But, if an expression within parentheses has a subtraction or negative sign before it, you must change the subtraction or negative sign, and you must change the sign of the quantities within the parentheses. That is, you make an addition problem out of a subtraction problem. Thus, 6 – (–8) becomes 6 + (+8). Here is another example: 6 – (+8) becomes 6 + (–8). If there is more than one quantity within the parentheses, you must change the sign of each quantity. For example, 6 – (–3 + 2) becomes 6 + 3 – 2, or 7. For this general situation, you can rewrite a – (b + c) as the formula a – b – c.
If you want to change the signs of expressions or numbers, you can reverse the process and put them within parentheses. For example, you can rewrite the expression 8 + 7 as –(–8 – 7). Or, you can rewrite 8 + 4 – 6 as 8 – (–4 + 6).
Fundamental laws.
There are five fundamental laws in algebra. These laws govern addition, subtraction, multiplication, and division. They are expressed in variables, and the variables can be replaced with any numbers. Here are the laws:
1. The Commutative Law of Addition is written x + y = y + x. This means that if you want to add two numbers, you can add them in either order, and the sum will be the same. For example, 2 + 3 = 3 + 2 = 5, and (–8) + (–36) = (–36) + (–8) = –44.
2. The Associative Law of Addition is written x + (y + z) = (x + y) + z. This means that if you want to add several numbers, you can add any combination first, and the final sum will be the same. For example, 2 + (3 + 4) = (2 + 3) + 4, or 2 + 7 = 5 + 4 = 9.
3. The Commutative Law of Multiplication is written x · y = y · x. This means that if you want to multiply two numbers, you can multiply them in either order, and the product will be the same. For example, (2)(3) = (3)(2) = 6, and (–8)(–36) = (–36)(–8) = 288.
4. The Associative Law of Multiplication is written x · (y · z) = (x · y) · z. This means that if you want to multiply several numbers, you can multiply any combination first, and the final product will be the same. For example, 2(3 · 4) = (2 · 3)4, or 2(12) = (6)4 = 24.
5. The Distributive Law of Multiplication over Addition is written x(y + z) = xy + xz. This law can be illustrated with an example: 3 · (4 + 5) = (3 · 4) + (3 · 5). If a number multiplies a sum, for example, 3(4 + 5), or 3 · 9, the result is the same as the sum of the separate products of the multiplier and each addend, (3 · 4) + (3 · 5), or 12 + 15. In this example, you can see that 3 · 9 = 12 + 15 = 27.
Other definitions.
It is important to define some other words used in algebra. An expression consisting of a product of numbers and variables is a monomial. For example, 5xy is a monomial. This particular monomial contains three elements (5, x, and y), called factors, that multiply each other. An expression with two or more terms connected by addition or subtraction symbols is called a polynomial. For example, x –y + z is a polynomial. One kind of polynomial is a binomial, an expression with two terms connected by an addition or subtraction symbol. For example, x + y and 3a2 – 4b are binomials.
A number, variable, or expression that acts as a multiplier is called a coefficient. For example, in the expression 5a, 5 is the coefficient of a and a is the coefficient of 5. In a(x + y), a is the coefficient of (x + y) and (x + y) is the coefficient of a.
Addition
in algebra is much like that in arithmetic. In algebra, a added to a is 2a. The expressions a and 2a are said to be like or similar because they contain exactly the same variables. To add two or more like quantities in algebra, you use the Distributive Law. In this way, 2x + 3x + 4x is (2 + 3 + 4)x, or 9x. But there is no single term for the sum of unlike quantities, such as a and b. This sum must be written a + b. To add 3a, 4b, 6a, and b, you can use the Commutative and Associative laws of addition. These laws permit you to add a series of numbers in any order. First, add the similar terms: 3a + 6a = 9a and 4b + b = 5b. Then, combine the sums. Thus, 3a + 4b + 6a + b = 9a + 5b.
You can use the following form to work out the problem:
To add unlike quantities that are both positive and negative, you can use the Distributive Law of Multiplication over Addition. The use of this law can be shown by adding (2a3 – b2c + 6bd2 + 2d3), (4a3 + 3b2c – 4bd2– 3d3), (3a3 + 2b2c + 2bd2 – 4d3), and (–2a3 – 8b2c + 6bd2 + 6d3). The little 3 above such terms as 2a3 means that the number represented by the variable is cubed. That is, the number is used as a factor three times (see Cube ). To add these terms, you should first arrange like terms in columns:
An explanation of the second column illustrates the method of addition. This column is –b2c + 3b2c + 2 b2c – 8b2c. Using the Distributive Law, you can see that these terms are the separate products of a multiplier, b2c. The coefficients are the numbers that make up a sum. These numbers are –1, +3, +2, and –8. You can add them together to obtain (–4)b2c, or –4b2c. Use the same method to add the other columns.
Subtraction
of products of numbers and variables follows the same rule as the subtraction of signed numbers. To subtract a number, you can simply reverse its sign and add it. In the example 8a – 3a, the sign of both the minuend and the subtrahend is positive. That is, (+8)a – (+3)a. Changing this from a subtraction problem to an addition problem converts it to (+8)a + (–3)a. The sum of 8a and – 3a is 5a.
The subtraction (2a3 – b2c + 6bd2 + 2d3) – (4a3 + 3b2c – 4bd2 – 3d3) can be handled as an addition problem with reversed signs. First, arrange like terms in columns:
Next, subtract the coefficients of like terms by changing the signsof the subtracted quantities and adding:
Multiplication
in algebra is usually indicated by writing two or more expressions together without an operation symbol. For example, a · b is written ab.
When a variable or number is multiplied by itself, the multiplication is abbreviated. For example, abb is written ab2 and abbbb is written ab4. The little number is called an exponent. It indicates the number of times a quantity is multiplied by itself. Thus, a · a or aa is written a2. It is called the square of a. Next, a · a · a or aaa is written a3. It is called the cube of a. And aaaa is written as a4, and aaaaa is written as a5. A variable that occurs by itself has an exponent of 1. If you are adding or subtracting exponents, you can write a as a1.
When you multiply like variables, you add their exponents. You can see that b2 · b3 is (b · b)(b · b · b), or b5. It is easier to add the exponents: b2 · b3 = b2 + 3, and b2 + 3 = b5. You cannot combine the exponents in a2 · b2 because a and b could possibly represent different numbers.
To multiply abcd by bc2dy, you combine the variables that are alike. In (abcd)(bc2dy), there are one a, two b’s or b · b, three c’s or c · c2, two d’s or d · d, and one y. So the product of abcd and bc2dy is ab2c3d2y. The Commutative Law of Multiplication permits you to multiply variables and numbers in any order.
To multiply an expression consisting of two or more terms by a single term or expression, you can use the Distributive Law of Multiplication over Addition: x(y + z) = xy + xz. Multiplying (3bd)(5b2c + 2d) shows the use of this law. You can modify the form used in arithmetic for multiplication:
To find the product in this example, you multiply the terms of (5b2c + 2d) one at a time. First, multiply 5b2c by 3bd. This product is 15b3cd. Write 15b3cd as the first term in the answer. Next, multiply 2d by 3bd. This product is 6bd2. Write 6bd2 as the second term in the answer. The total product is 15b3cd + 6bd2.
To multiply two expressions each consisting of two or more quantities is more difficult. Here is an example. The problem is (a2– 2ab + b2)(a – b).
First, multiply each term in the top expression (a2 – 2ab + b2) by a, the first term of the bottom expression (a – b). Write the product of this multiplication.
Next, multiply each term in the top expression by b, the second term of bottom expression. Write this second product below the first product. Arrange like terms in columns.
Last, add the two products to obtain the total product. Arranging like terms in columns helps you to do the addition that gives the total product.
Division
in algebra is the opposite of multiplication. Remember that to multiply like terms, you add their exponents. To divide like terms, you subtract the exponents. For example, b5 ÷ b2 = b5 – 2 and b5 – 2 = b3.
Here is a more difficult problem: (3x4y2z – 9x3yz2 – 6x2y3) ÷ (3x2y). In this case, you must divide each part of the first expression in turn by (3x2y). For each part, ask what multiplied by (3x2y) will give that part of the dividend. For example, what quantity, when multiplied by (3x2y) will give (3x 4y 2z) ? The answer is (x 2yz). . Using this method, (3x4y2z – 9x3yz2 – 6x2y 3) ÷ (3x2y) = (x2yz – 3xz2 – 2y2).
Here is another problem: (12a2 + 18ab + 6b2) ÷ (4a + 2b). For a problem of this kind, you can use a form somewhat like the form used in arithmetic for long division:
First, divide the first term of the dividend by the first term of the divisor: 12a2 ÷ 4a = 3a. Write the result, 3a, as the first term in the quotient to the right. Next, multiply both terms of the divisor by 3a, the first term in the quotient: (4a + 2b)(3a) = 12a2 + 6ab. Write this product below the dividend and subtract it from the dividend. You must account for the result of this subtraction, 12ab + 6b2, with a second term in the quotient. To do this, divide 12ab by the first term of the divisor: 12ab ÷ 4a = 3b. Multiply the divisor by 3b: (4a + 2b)(3b) = 12ab + 6b2. You can see that there is no remainder.
Factoring
means to find expressions that are factors of a given product. For example, (4a + 2b) and (3a + 3b) are factors of 12a2 + 18ab + 6b2. If you multiply (4a + 2b)(3a + 3b), the product is 12a2 + 18ab + 6b2. An expression can have more than one set of factors. For example, 2 X 12, 3 X 8, and 4 X 6 are sets of factors of 24. Factoring is important in algebra because it is used to simplify complicated expressions (see Factor ).
Working with equations
Functions.
The amount of gasoline used by an airplane is related to its speed. The amount of postage required for a parcel depends on its weight. The idea of one thing depending on another is important in mathematics. It is called the relation of one thing to another. In algebra, a certain relation of two variables is called a function.
You can learn the idea of a function from familiar things. For example, imagine a concrete foundation that is 16 centimeters above the level of the ground. On this foundation, you build up 6 layers of stone blocks. Each layer is 8 centimeters thick. As you add each layer of blocks, the height of the pile becomes larger.
Use x to represent the number of layers and y to represent the height of the pile. Here is a table showing the relation of the number of layers of stone blocks to the height of the pile.
You can show the numbers in this table on a graph. Distances along two lines represent the values of x and y. One line is horizontal and shows values of x. The other line is vertical and shows values of y. These two lines are called coordinates. You can plot the number pairs from the table on the graph with 7 dots.
There is an equation that describes this line of dots: y = 8x + 16. You can see how the equation fits the table of values. For example, if x = 2, then y = 8(2) + 16, or 32. If x = 5, then y = 8(5) + 16, or 56.
In the equation, the domain of x is the set of numbers {0,1,2,3,4,5,6}. The values of y are called the range of y. The range of y is the set of numbers {16,24,32,40,48,56,64}. Mathematicians call the relation between the two sets of numbers a set of ordered pairs. This set is written {(x,y)}→{(0,16), (1,24), (2,32) … (6,64)}. This set of pairs is a function. It is called a discrete function because it cannot be represented by a continuous line.
Now, imagine that the bottom of an aquarium is 20 inches above the floor. The aquarium is 36 inches high. Water flowing into the aquarium causes the level of the water in the aquarium to rise 4 inches every minute.
This means that the height of the water above the floor is related to the time the water has been flowing. In this example, use x to represent the number of minutes the water has been flowing and y to represent the distance of the surface of the water from the floor. Here are some of the values of x and y:
When this relation is shown on a graph, the line is solid because the height of y increases continuously. You can describe this line with an equation: y = 4x + 20. If x = 2, then y = 4(2) + 20, or 28. You can see how this equation fits the table of values. The domain of x is all numbers between 0 and 9, and the range of y is all numbers between 20 and 56. This function is called a linear function because it is continuous and can be represented by a solid line. The equation y = 4x + 20 is called a linear equation. The study of linear equations is one of the most important topics in algebra.
Solving linear equations in two variables.
The equation y = 4x + 20 is linear. It has two variables, x and y. Every point on this line represents two numbers that count as a solution to the equation. Thus, many pairs of numbers make y = 4x + 20 a true statement.
Because linear equations have many solutions, it is often useful to find some sort of restriction or limit for them. For example, you might want to use a linear equation to solve a practical problem. To do this, you must find some way to restrict the equation to one set of values. One method is to use a pair of equations that are true for only one pair of numbers.
The equations 2y = x + 4 and y + x = 5 illustrate this method. To solve these equations, you can use a graph. First, make tables of a few of the values that solve each equation.
Plot these points on the graph and draw a line for each equation. The two lines cross. The point where they cross represents the numbers that will solve both equations. This point is (2, 3). That is, x has the value 2, and y has the value 3. Only these values for x and y will solve the two equations.
You can also solve a pair of linear equations in two variables by eliminating one of the variables. This results in a single equation in one variable. You can use 2y = x + 4 and y + x = 5 as examples. There are various ways of eliminating a variable. The method that can be used here is called substitution. First, solve one of the equations for y. That is, find what y equals in one of the equations. Naturally, you will not know the value of x. Using the equation y + x = 5, the solution for y is 5 – x. Substitute this for y in the other equation. The other equation is 2y = x + 4. Substituting the new value for y, you obtain 2(5 – x) = x + 4. Simplified, this is 10 – 2x = x + 4. Subtracting 4 from both sides of this equation produces 6 – 2x = x. Adding 2x to both sides of the new equation produces 6 = 3x. Therefore, x = 2. Now you can replace x with 2 in either of the two equations and find 3 as the value for y: 2y = 2 + 4 and y + 2 = 5. So the solution of the two equations is {(2,3)}.
An equation in two variables can also be solved by restricting the solution to positive whole numbers. You can see this in a problem involving a man who bought some prize turkeys and ducks. He spent $31. He paid $5 for each turkey and $2 for each duck. How many of each did he buy? Use x to represent the number of turkeys and 5x as their cost. Use y to represent the number of ducks and 2y as their cost. You can write this problem as an equation: 5x + 2y = 31. You can substitute only whole numbers for x and y because, in this case, you cannot buy part of a bird.
To solve this problem, we use the fact that only an even and an odd number can have the sum 31. Any whole number multiplied by 2 is an even number, so 2y is even. This means that 5x must be an odd number. Any even number multiplied by 5 is an even number, so x cannot be even. Any odd number multiplied by 5 is an odd number, so x might be any odd number. Replacing x with odd numbers, you will find that the pairs of numbers that solve the equation are (1,13), (3,8), and (5,3). The man could have bought 1 turkey and 13 ducks:
Or three turkeys and eight ducks:
Or five turkeys and three ducks:
You cannot use 7 for x because the pair would be (7,–2), and –2 ducks is not a solution.
For another method of solving equations in two variables, see Determinant .
Quadratic equations in one unknown.
A quadratic equation is one in which the variable is squared. For example, x2 – 8x = –16 is a quadratic equation in one unknown. By combining terms, you can put any quadratic equation with one unknown variable in the following form:
ax2 + bx + c = 0
In this formula, a, b, and c represent the known numbers or coefficients. For example, b is the coefficient of x, and x is the unknown variable. In the simplest example of this kind of equation, the coefficient a = 1 and the coefficient b = 0. For instance, if a = 1, b = 0, and c = –36, then x2 – 36 = 0. This means that x2 = 36 and the solution set is {6,–6}. If b does not equal zero, there are three other methods for solving this type of equation.
The first method is to factor the equation after it has been put in the form ax2 + bx + c = 0. You can use x2 + 8x + 15 = 0 as an example. You can factor the left-hand side of this equation: x2 + 8x + 15 = (x + 3)(x + 5). So (x + 3)(x + 5) = 0. If the product of two numbers is zero, one of the numbers must be zero. If x + 5 = 0, then x = –5. Similarly, if x + 3 = 0, then x = –3. The solution set of x2 + 8x + 15 is {–3,–5}.
The second method is called completing the square. An expression such as a2 + 2ab + b2 is called a perfect square because it can be rewritten (a + b)2. You can change an equation such as x2 + 8x + 15 = 0 so that the left-hand member is a perfect square. To do this, rewrite the equation x2 + 8x + 15 = 0 as x2 + 8x = –15. You know that x2 + 8x + 16 is a perfect square because it can be rewritten (x + 4)2. You can add 16 to both sides of x2 + 8x = –15. This gives you x2 + 8x + 16 = –15 + 16. Factoring this, you find that (x + 4)2 = 1. One of two equal factors of a number is called its square root (see Square root ). In the equation (x + 4)2 = 1, x + 4 must equal the square root of 1. The square root of 1 is either 1 or –1. So x + 4 = 1 or x + 4 = –1. Then x = –3 or x = –5. This means that the solution set to x2 + 8x + 15 = 0 is {–3,–5}.
The third method for solving a quadratic equation in one unknown is to use the following formula:
You can obtain the coefficients a, b, and c from any quadratic equation put in the form ax2 + bx + c = 0. Substituting these numbers in the formula will give you the value of x. In the formula, the symbol ± means positive or negative. It also indicates that there will be two roots to the equation.
Here is how to apply the equation to the formula x2 + 8x + 15 = 0:
Therefore, the solution set to x2 + 8x + 15 = 0 is {–3,–5}.
History
The ancient Egyptians and Babylonians used algebra. Evidence for its development appears in an Egyptian book that was copied by the scribe Ahmes about 1650 B.C. The Babylonians used more advanced algebra than did the Egyptians. Hundreds of years later, the Greeks, Chinese, and people of India contributed to the development of algebra. Diophantus, a mathematician who lived in the A.D. 200’s, used quadratic equations and symbols for unknown quantities.
The Arabs made many contributions to the study of algebra. They adopted the number system of India, including the zero, and developed fractions much as they are used today. They helped transmit earlier mathematical ideas to the West. In the early 800’s, the Persian mathematician al-Khwarizmi wrote an influential book on algebra that was used as a textbook. Most scholars consider either al-Khwarizmi or Diophantus to be the father of algebra. The English word algebra comes from an Arabic word meaning restoration or completion in the title of this work. The Persian astronomer and poet Omar Khayyam (c. 1048-1131) wrote a book on algebra.
There was little progress in the development of algebra during the Middle Ages. Europeans began to study the subject in the late 1400’s and in the 1500’s. Many mathematicians contributed to its later development.
The widespread use of computers has caused major changes in the study and use of algebra. Inexpensive software can perform most problem-solving steps studied in algebra. For example, the programs can quickly solve linear or quadratic equations. The emphasis in algebra classes has therefore begun to shift from learning basic symbol-manipulation skills to understanding algebra’s underlying concepts.
