EXAMPLE 1. Solve the equation 7 x+27=3 x–9. (Given) Subtracting 3 x from both sides, we have 4x+27=-9. (Axiom II, § 9) Subtracting 27 from both sides of the last equation, we find 4x=-36. (Axiom II) Dividing both sides by 4, we have x=-9. Ans. (Axiom IV) CHECK. With x=-9 the right side of the given equation becomes 7(-9)+27, or – 63+27=-36. At the same time its left side becomes 3(-9)–9, or – 27–9=-36. Since both sides thus become the same (namely, -36), the answer must be correct; that is, -9 is the desired value of x. EXAMPLE 2. Solve the equation 3 x–9=8*x+20. (Given) Subtracting 8 x from both sides, we have -5 x-9=20. (Axiom II) Adding 9 to both sides, we find -5 x=29. (Axiom I) Dividing both sides by -5, we have x=-54. Ans. (Axiom IV) EXERCISES Solve the following and check your answer for each. 1. x+8=10. 4. 6— x=3. 7. 4-X=-3. 2. x+8=7. 5. –4+x=8. 8. —x+5=24. 3. 4-x=7. 6. –4+x=-8. 9. x+1=-1. 10. 8x+2 x= -12. 14. 4 x=6 x-22. 11. 4 x–9 x=25. 15. 3 x-10=20. 12. 10-2 x=4. 16. 2 x+16=8x+4. 13. 2 x–18=4 x. 17. 13 x—4=18 x+14. 18. 7 K+36= -2 K-90. 23. 7b+13=43–2 b. 19. – 50 r+2 r=100+77 r. 24. –3 a+b=a+18. 20. 8y-16=3 y+30. 25. – 10 r+15= –25. 21. 6z+15–4z=21+32–8. 26. 64 x–8+24 x=-17. 22. 12 k-9+32= 24 k-13+ k. 27. 2.3 r-4.6=1.2 +3.2. 28. The sum of two numbers is 12 and one of them is twice the other. Find the numbers. (Hint. Let x=the smaller number. Then 12–x=the larger number.] 29. The sum of two numbers is 12 and their difference is 14. Find the numbers. 30. If a certain number be subtracted from 12 the remainder is 19. What is the number? 31. If 8 be subtracted from four times a certain number, the result is 16 more than twice the number. Find the number. 32. The sum of three numbers is 21. The second is 6 more than the first, and the third is 2 less than the first. Find the numbers. [Hint. Let x = the first number.] 33. The sum of three numbers is 21. The second is 15 more than the first, and the third is 19 less than the second. Find the numbers. 34. The length of a certain rectangle is 4 feet more than twice the width. The whole distance around (called perimeter) is 56 feet. What is the length and what the width ? 35. A rectangle whose perimeter (see Ex. 34) is 98 feet is 18 feet longer than wide. Find its dimensions (length and breadth). 36. Find two consecutive numbers whose sum is 17. [HINT. Two numbers are said to be consecutive when the one is 1 greater than the other. Thus, 2, 3 are consecutive; so also are 10,11; so also are -3, -2. More generally, several numbers are consecutive when each is 1 greater than the one before it. Thus, 2, 3, 4, 5, 6 are consecutive; so also are -2, -1,0, 1, 2, 3, 4; so also are }, }, }, 10.] 37. Find three consecutive numbers such that their sum is equal to the last number increased by 17. (See Hint to Ex. 36.) 38. Find four consecutive numbers such that the sum of all four of them is the same as the sum of the first and last. For further exercises on this topic, see Appendix, pp. 293–295. CHAPTER IV PARENTHESES 38. Definition. Parentheses ( ) are used to show that the terms included within them are to be regarded as one number. Thus, 6+(3-2) means that we are to add 3-2 or 1, to 6. That is, 6+(3-2)=6+1=7. Ans. Similarly, 6– (3-2) means that we are to subtract 3-2, or 1, from 6. That is, 6-(3-2)=6–1=5. Ans. Other examples, which should be carefully examined, follow. (2+3)+(2-1)=5+1=6. Ans. 4-(5-7)=4-(-2)=4+2=6. Ans. (a+2)+(a-3)=a+2+a-3=2 a-1. Ans. 39. To Multiply a Quantity Inclosed in Parentheses. When a number is placed directly before or after an expression inclosed in parentheses, with no sign between them, multiplication is indicated. Thus, 2(4-2) means 2X(4-2); that is, 2X2, or 4. Similarly, 3(a+5) means that the sum of a and 5 is multiplied by 3. Again, (a+2)(a–5) means that the sum of a and 2 is multiplied by the difference between a and 5. ORAL EXERCISES 1. Give the value of each of the following expressions : (a) 4+(3-1). (e) 8(4-3). (6) 4-(3-1). (f) (2-3)(4-5). (c) (2+3)+(5-6). (g) 7(6-1)(4-5). (d) (2+3)–(1–2)+(3+4). (h) 2(3-1)+(6-7). 2. Read each of the following expressions : (a) 5(a+2). (d) 4(a? — 2)(a? — 1). (6) 6(a−b). (e) (a-1)(a? — 2)(a3–3). (c) (a+3)(a+2). (f) (a−b)2 (a−6+2). WRITTEN EXERCISES 1. Find the value of each of the parts of Ex. 2 in the last set of exercises when a=1 and b=2. 2. Write out the algebraic expression for each of the following phrases : (a) Four times the sum of r and s. (f) The sum of x square and y square multiplied by the sum of a and b. (g) The sum of x square, y square, and z square multiplied by the difference between r cube and s cube. (h) Seven times the difference between x square and y square multiplied by the sum of f, g, and h. (c) The sum of the squares of a, b, and c multiplied by the sum of the cubes of x, y, z, and w. (j) The cube of the difference between a and b. (k) The cube of the sum of a, b, and c. 40. Removing Parentheses. Such an expression as 10+(6+2), in which the sign before the parentheses is +, may be simplified in either of two ways: (1) By getting the value of the part in parentheses and adding it to 10. Thus, 10+(6+2)=10+8=18. |