ELEMENTARY ALGEBRA Lesson No. 1. Introductory T 'HE elementary processes of algebra differ but slightly from those of arithmetic. In arithmetic numbers are represented by figures, the values of which are known. In algebra numbers are represented largely by letters which have an unknown or an unassigned value. The following arithmetical symbols, with some others which will be introduced later, are used in algebra: + plus, the sign of addition. -minus, the sign of subtraction. × multiplied by. + divided by. ✔square root of. = is equal to. When two figures are written alongside of each other, the first figure represents tens, and the second units. In algebra, when two or more letters are written alongside of each other, it indicates that the numbers represented by the letters are to be multiplied together. Thus: In the last illustration the 3 is called a coefficient. When the coefficient is 1, it is omitted. If we want to indicate a+a+a in algebra, we write 3 a, which means 3 times a. Thus in 3 abc the 3 denotes that the product of a, b, and c is to be taken 3 times. An algebraic expression is a quantity expressed in algebraic language. is an algebraical expression. This expression is read as follows: Two ab, plus three xy, minus six m, plus five. The expression Two a square, plus ab, minus three b cube. In the second illustration the 6 a is to be multiplied by the sum of a and b. The brackets indicate that a + b represents one quantity, and this quantity is to be multiplied by 6a. In the third illustration the 2 a2 (read "two a square") is equal to 2 x ax a. Instead of writing a twice, or three times, or four times, we write a small number called an exponent a little to the right and above the number. 363 (read "three b cube") is equal 3 xbx b xb. EXERCISES When a = 1, b = 2, c = 3, d = 4, e = 5, find the numerical value of the following algebraic expressions: 1. 4a+3b+2c-d. 6. 6a (b+c+d) - 3 a. 7. 5(b+c)-2 a (d2 — e). 8. 3 abc-c+ b (3b+4). 9. (a+b)2+3 b (15 — c). 10. (a2 + b2) b + 2 abc. Lesson No. 2. Algebraic Addition An algebraic expression whose parts are not separated by + or is called a term. For example, in each of the following expressions there are three terms: NOTE. In the second of the above expressions, though the expression consists of three terms, the first term is in part made up of two terms. Similarly, in the third of the expressions, the second term is itself made up of two terms. A term which has the sign plus before it is called a positive term. When the first term of an expression is positive, the sign plus is usually omitted. A term which has the sign minus before it is called a negative term. The sign is part of the term, and must be moved about with it if the position of the term in the expression is changed. Terms which contain the same letters and exponents are called similar terms. The coefficients and signs may be different, and yet the terms be similar. In the following expressions the terms are similar in each expression: 3ab4abab + 6 ab, 2 a2bc-12 a2bc + 3 a2bc. As in arithmetic we can combine—that is, add or subtract-only like quantities, so in algebra. Thus: 5 ft.7 ft. 12 ft., 5 ab +7 ab = 12 ab. Five feet plus seven inches could be expressed 5 ft.+ 7 in.; so in algebra 5 a plus 7 b is expressed 5 a +76. Note the following: 12-15, impossible in arithmetic; In algebra we have enlarged our ideas of numbers to include 1,2,3,4, etc., as well as 1, 2, 3, 4, etc. other words, we count in both directions. In the illustration, if 12 represented 12 lb. pulling in a plus direction, and 15 represented 15 lb. pulling in a minus direction, the difference would be 3 in the minus direction, or -3. Thus: 5b-7b - 2b, -8a3a=-5a, -6b-3b-9b. To add similar terms you find the difference between the sum of the positive terms and the sum of the negative terms, giving the result the sign of the larger sum. EXERCISES Write the following expressions in their simplest form : 1. 8a +7a+ 12a + 3 a + 16 a. 2. 2a 3a-7a-9a-15 a. Lesson No. 3. Simple Equations An equation is a statement of equality between two expressions. Thus : x+3x+5x= 27. The parts of an equation to the right and left of the sign of equality are called the members or sides of the equation, and are distinguished as the right side and left side. |