Elementary Algebra and Geometry for Schools

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 HOME : Mathematics - Geometry Literal Numbers; Evaluating Formulas Literal numbers. In your study of arithmetic you dealt with numbers composed of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and combinations of these digits, such as 17, 509, 2/3, 6.4, and √83. In algebra, you will continue to use these symbols for numbers, but you will also use letters to represent numbers. Thus a will mean the number represented by the letter a; b will mean the number represented by the letter b; and x will mean the number represented by the letter x. This use of letters to represent numbers is not entirely new to you. You will remember that in arithmetic you found the area of a rectangle by multiplying its length by its width. You probably expressed this fact by the formula A = l X w, where A represented the number of square units of area in the rectangle, l represented the number of linear units in its length, and w represented the number of linear units in its width. Thus in your study of formulas in arithmetic, you became familiar with literal numbers, that is, with numbers which are represented by letters. Literal numbers are numbers which are represented by letters. Evaluating a formula. In the formula A = l X w, the numerical value of A can be found when the numerical values of land ware given. Finding the value of A for certain given values of land w is what we mean by "evaluating the formula." In evaluating this formula, if l and w are expressed in inches, then A is expressed in square inches; if l and w are expressed in feet, then A is expressed in square feet; if l and w are expressed in yards, then A is expressed in square yards. ILLUSTRATIVE EXAMPLE: Evaluating a formula Find the area of a rectangle whose length is 6 in. and whose width is 4 in. Solution. In the formula A = l X w, we substitute 6 for l and 4 for w, thus A = 6 X 4, A= 24 Hence the area of the rectangle is 24 sq. in. EXERCISES: Evaluating a formula In the following problems, find the value of A in the formula A = l X w, when the values of l and w are: Answers : 1. 15 sq. in. ;2. 135 sq. ft. ;3. 25.2 sq. mi. ;4. 190.4 sq. yd. ;5. 46.8 sq. in. ;6. 29.52 sq. yd. ;7. 26 sq. ft. ;8. 52 sq. in, ;9. 2/5 sq. in. ;10. 14 sq. ft. Indicating addition and subtraction. The symbols for addition and subtraction, + (plus) and - (minus), are the same in algebra as in arithmetic. In arithmetic, to indicate that 7 is to be added to 2, we can write 2 + 7; in algebra, to indicate that some number represented by the letter b is to be added to some other number represented by the letter a, we can write a + b. In arithmetic, to indicate that 5 is to be subtracted from 8, we can write 8 -5; in algebra, to indicate that some number d is to be subtracted from some other number c, we can write c -d. In arithmetic where we are dealing with specific numbers, the indicated addition or subtraction of the numbers can be completed, that is, 7 + 5 equals 12, or 35 -17 equals 18. In algebra, when one or more than one of the numbers are represented by letters, the addition or subtraction can be indicated only; that is, the addition or subtraction cannot be completed unless the numerical value of each of the letters is given. To indicate that x is to be added to y, we write: x +y or y+x. To indicate that 7 is to be subtracted from a, we write: a -7. ILLUSTRATIVE EXAMPLES: Expressing words in algebraic symbols. 1. Express in algebraic symbols, "the sum of c and d." Solution. c +d 2. Express in algebraic symbols, "5 less than b." Solution. b -5 EXERCISES: Expressing words in algebraic symbols Express each of the following in algebraic symbols: Answers : 1. a + b. ;2. x -y. ;3. a-b. ;4. y -3 ;5. x + a. ; 6. x -5. ;7. 6+ a. ;8. x + y. ;9. r + s. ;10. 7 - x. ;11. c + 7. ;12. d -8. ;13. x + y + z. ;14. m -n. ;15. 33 + a. ;16. x + b. ; 17. y -15. ;18. x - 87. ; 19. r + s. ;20. y - x. ; 21. a-b. Indicating multiplication. The times sign, X, that is used in arithmetic to indicate multiplication is used only rarely in algebra. This is because the symbol x, when written, may be confused with the letter x, and the letter x is frequently used in algebra to represent a number. In algebra, a raised dot, • , may be placed between numbers, or between letters that represent numbers, to indicate multiplication. Thus  6•7 indicates that 6 is to be multiplied by 7; c•d indicates that c is to be multiplied by d; and 8•y indicates that 8 is to be multiplied by y. When you use the raised dot to indicate multiplication, you must be sure that it is raised so that there is no confusion between this symbol and a decimal point; that is, 9•4 means nine times four; whereas, 9.4 means the number, nine and four tenths. If there is no possibility of confusion, the raised dot may be omitted; that is, two letters written next to each other indicate that the numbers represented by those letters are to be multiplied. Thus xy indicates that x is to be multiplied by y. Also, a number and one or more letters written in succession indicate that the number and the numbers represented by the letters are to be multiplied. Thus 2a indicates that 2 is to be multiplied by a; and 7cd indicates that 7, c, and d are to be multiplied. The raised dot may not be omitted between two numbers. That is, 3•7 indicates that 3 is to be multiplied by 7, whereas 37 means the number thirty-seven. To indicate that 8 is to be multiplied by a, write 8•a or 8a. Indicating division. Division may be indicated in algebra in the same ways in which division is indicated in arithmetic, namely, by the division sign,÷, and by a fraction whose numerator is the dividend and whose denominator is the divisor. To indicate that x is to be divided by y, write x ÷ y or or x/y. Indicating multiplication by a fraction. The multiplication of a number, numerical or literal, by a fraction is the same in algebra as in arithmetic. The number is multiplied by the numerator of the fraction, and this product is divided by the denominator of the fraction. We use the same symbols to indicate multiplication of a number by a fraction as we used to indicate multiplication of whole numbers and literal numbers. Thus, to indicate that the fraction 2/3 is to be multiplied by a, we may write 2/3•a, or 2/3a. Since both of these mean that 2 is to be multiplied by a and this result divided by 3, we may also write 2a/3. In indicating that 3 is to be multiplied by ½, we may use the raised dot, or the multiplication sign X; we must not write the 3 and ½ the next to each other, with no symbol between them, to indicate multiplication, since 3½ indicates 3 plus ½, just as it does in arithmetic. ILLUSTRATIVE EXAMPLES: Expressing words in algebraic symbols 1. Express in algebraic symbols, "the product of 9 and x," Solution., The product of two numbers is the result obtained by multiplying the two numbers together. We may indicate the product of 9 and x, in simplest form, as 9x 2. Express in algebraic symbols, "three fifths of b." Solution. This means that we are to multiply 3/5 by b. We may indicate this product, in simplest form, as EXERCISES: Expressing words in algebraic symbols Express each of the following in algebraic symbols: Answers : ORAL EXERCISES In which of the following may the X sign be omitted? (You do not need to find the products.) An algebraic expression. Whenever numbers, or letters which represent numbers, are combined with the symbols that indicate addition, subtration, multiplication, or division, an algebraic expression is formed. Thus; are algebraic expressions. In the preceding exercises you learned to express a verbal statement, such as "the sum of x and y," as an algebraic expression, x + y; or a verbal statement, such as "the product of a and b," as an algebraic expression ab . Algebraic expressions such as x + y, lwh, and x/y can be evaluated when the value of each of the letters is known. ILLUSTRATIVE EXAMPLES: Evaluating algebraic expressions  EXERCISES: Evaluating algebraic expressions If a = 3, b = 7, and c = 15, find the value of each of the following: Answers : 1. 18. ;2. 9. ;3. 3. ;4. 8. ;5. 22. ;6. 30. ;7. 3. ;8. 21. ;9. 4. ;10. 1 ;11. 14. ;12. 6. ;13. 0 ;14. 25 ;15. 105 ;16. 5 ;17. 150 ;18. 0 ;19. 45 ;20. 1 ;21. 5 ;22. 42 ;23. 16 ;24. 0 :25. 5/3 :26. 1 ;27. 2 ;28. 10 ;29. 3/2 ;30. 5 ;31. 9 ;32. 11 ;33. 2 ;34. 4,15,10,20,3,7. ;35. 18,16,48,192,2,14 ;36. 50,2,15,5, 100,15 ;37. 32,3,4,20,16, 8½. 38. 0,0,2,1,3. Formulas. In your work in arithmetic you have already become familiar with many different formulas. You have seen how with a few symbols a formula may express a principle. The same principle, stated as a rule, might require many words. You know that "the volume of a rectangular solid (a block) is found by multiplying the length by the width by the height." This statement can be expressed by the formula: V = lwh A formula is algebraic shorthand. where V represents the number of cubic units in the volume, I represents the number of units in the length, w represents the number of units in the width, and h represents the number of units in the height. If we are given the values of l, w, and h, we can substitute these values into the formula and find the value of V by performing the indicated operations. ILLUSTRATIVE EXAMPLE: Evaluating a formula In the formula V = lwh, find the value of V if l = 8 in., w = 5 in., and h = 3 in. Solution. Given formula V = lwh. Substituting 8 in. for l, 5 in. for w, and 3 in. for h, we obtain V = 8.5.3 Simplifying V = 120 cu. in. Answer EXERCISES: Evaluating formulas 1. The formula for the perimeter of a triangle is P = a + b + c, where P is the perimeter of the triangle, and a, b, and c are the lengths of the sides of the triangle. In each of the problems below, you are given the numerical values of a, b, and c in inches, and you are to find the numerical value of P in inches, by evaluating the formula P = a + b + c.  1. (a) 22; (b) 61; (c) 12 3/4; (d) 13.5. 2. The formula for the volume of a rectangular solid is V = lwh, where V is the volume of the solid, l is the length of the base, w is the width of the base, and h is the height. In each of the problems below you are given the numerical values of l, w, and h in feet and you are to find the numerical value of V in cubic feet by evaluating the formula V = luh.  2. (a) 900; (b) 240; (c) 15; (d) 44.1. 3. The formula for the distance traveled by a body at a constant rate of speed is D = rt, where D is the distance traveled by the body, r is the rate of speed, and t is the time the body is in motion. In each of the problems below you are given the numerical value of r in miles per hour and of t in hours, and you are to find the numerical value of D in miles hy evaluating the formula D = rt. 3. (a) 240; (b) 275; (c) 270; (d) 15.75. 4. The formula for the area of a triangle is A = ½bh, where A is the area of the triangle, b is the length of the base of the triangle, and h is the length of the altitude of the triangle. In each of the problems below you are given the numerical values of band h in feet, and you are to find the numerical value of A in square feet by evaluating the formula A = ½bh  4. (a) 30; (b) 207; (c) 50; (d) 50.05. 5. The formula for the perimeter of a quadrilateral (a four-sided figure) is P = a + b + c + d, where P is the perimeter and a, b, c, and d are the lengths of the sides of the quadrilateral. In each of the problems below you are given the numerical values of a, b, c, and d in inches and you are to find the numerical value of P in inches by evaluating the formula P = a + b + c +d.  5. (a) 30; (b) 46; (c) 16; (el) 30.4. 6. The formula for the circumference of a circle is C = 2πr, where C is the circumference of the circle, π is a number approximately equal to 3.1416, and r is the radius of the circle. In each of the problems below you are given numerical values of π and r, and you are to find the numerical value of C by evaluating the formula C = 2πr. 6. (a) 88; (b) 220; (c) 124; (d) 31,416. The origin of numbers. Mathematics had its beginning when men first learned to count. They probably began by counting on their fingers and, since there are ten fingers on the two hands, the number 10 became the basis of most number systems. When men learned how to write, the number 1 was written as a short vertical stroke, representing one finger. Similarly, the number 2 was written as two short vertical strokes, representing two fingers. The number 3 was written as three strokes, and so on up to nine. To represent 10, the ancient Egyptians used the symbol ∩. Simple equations. An equation is an algebraic statement of equality. The formulas we have used in arithmetic and in this chapter are equations. The formula A = lw states in algebraic shorthand that the area of a rectangle equals the product of its length multiplied by its width. An equation may also ask a question. The equation 20 = 5w asks, "Twenty equals 5 multiplied by what number?" ILLUSTRATIVE EXAMPLE: Solving simple equations 1. Solve for w: 12=6w Solution. Given equation 12 = 6w This means " 12 equals 6 multiplied by what number?" We write 12 = 6. ? Since 12 equals 6 times 2, we write 12 = 6 .2 Thus w = 2 Answer 2. Solve for a: 5a = 30. Solution. Given equation 5a = 30 This means" 5 times the number represented by a equals 30." We write 5 .? = 30 Since 5 times 6 equals 30, we write 5 . 6 = 30 Thus a = 6 Answer 3. Solve for m: ½m = 10. Solution. Given equation ½m = 10 This means "One half of the number represented by m equals 10." We write ½· ? = 10 Since ½ of 20 equals 10, we write ½· 20 = 10 Thus m = 20 Answer EXERCISES: Solving equations Find the value of the literal number in each equation: 1. w = 8 ;2. l = 6. 3 ;a = 12. 4 ;y = 2. 6 ;a = 6 ;6. b = 9. 7. n = 40 ;8. c=12 ;9. n = 12 ;10. y = 7 ;11. b = 8 ;12. x = 2 ;13. x = ½ ;14. x = ¼ ;16. x = 6. REVIEW EXERCISES 1. 102 sq. in. ; 2. 9.8. ; 3. 70 ; 4. x + y + z ; 5. ab ;  6. 2ab ; 7. b and d ; 8. no ; 9. yes ; 10. V = 210 ; 11. A = 30 ; 12. ¾y ; 13. (a) n = 7; (b) n = 9 ; 14. (a) x = 24; (b) x = ½