59. Sometimes it is advisable to simplify in the course of the work. Example. Find the value of 84-7[-11.0-4{-17.3 +3 (8-9 - 5x)}]. =84–7[- 11x – 4{-1730+3 (5x - 1)}] When the beginner has had a little practice the number of steps may be considerably diminished. EXAMPLES VII. b. Simplify by removing brackets 1. a - [2b+{30 - 3a -(a+b)) + 2a -(1+30)]. 2. a+b-(c+a-[b+c-(a+b-{c+a-(6+c-- a)})]). 3. a-(6-c)-[a-b-c-26+c-3(c-a)-d]. 4. 2.x -(3y - 42) - (2x - (3y+42)} - {37-(42+2.c)). 5. b+c-(a+b-[c+a-(b+c-{a+b-(c+a-6)})]). 6. 36– {5a - [6a+2(10a -b)]}. 7. a-(6-c)-[a-b-c-2{b+c]. 8. 3a2-6a? - {862-(9c2 - 2a)]. 9. b-c-a) - [b-a-c-2{c+a-3 (a -1)-1]. 10. — 20 (a -d)+3(6 - c) – 2[b+c+d– 3{c+d-4(el – a);]. 11. - 4(a+d)+24 (-c) - 2[c+d+a-3d+a - 4(b+c)}]. 12. - 10(a+b)-[c+a+b-3{a+26-(c+a -- 0);]+4c. 13. a-2(6–c)-[-{-(4a-1-c- 2 {a+b+c}}}]. 14. 8(6 - c)-[-{a-6-3 (c-b+a)}]. 15. 2 (36 - 5a) -7[a-6 2-5(a-)]. 16. 6{a-2[5-3(c+d)]}-4{a-3[b – 4(0-d)]}. 17. 5a-2[a-2(a+x)]}-4a-2[a-2(a+x)]). 18. - 10{a-6[a-(6-c)]}+60{6-(c+a)). 19.-31-2[-4(-a)]}+5{-2[-2(-a)]}. 20. -%{-[-(-y)]}+{-2[-(x - y)}}. 3/1 2 3 -56} b 2 3 3x – 44 22. 35 - C 23.3{(a - b)-8(-c)} Sb - }{c-a-3 (a-6)}. 2 24. 3x – }(šy – }z) – [x – {}x – (3y – $2)} - (y – 57)]. INSERTION OF BRACKETS. 60. The converse operation of inserting brackets is important. The rules for doing this have been enunciated in Arts. 21, 22; for convenience we repeat them. 1. Any part of an expression may be enclosed within brackets and the sign + prefixed, the sign of every term within the brackets remaining unaltered. 2. Any part of an expression may be enclosed within brackets and the sign prefixed, provided the sign of every term within the brackets be changed. Examples. a-b+c-d-era – 6+(c-d-e). a-b+c-d-era-(6-c) – (d+e). wy – a:c – by +ab=(xy – by) – (ax - ab). 61. The terms of an expression can be bracketed in various ways. Example. The expression ax – bx+cx – ay+by - cy may be written (ax – bx) +(cx – ay) + (by - cy), (ax - ay) - (x-by)+(cx - cy). 62. A factor, common to every term within a bracket, may be removed and placed outside as a multiplier of the expression within the bracket. or or Example 1. In the expression ax? – cx + 7 - do? + bx - C - dxc3 + 0.x2 – 2x bracket together the powers of x so as to have the sign + before each bracket. The expression=(axc3 – dor:3) + (bxc2 – dx2) + (bx – cx – 2x0) +(7 — c) =(a – d) + ac* (6 – d) + x (6-c-2)+(7 — c) =ía – d) x3 + (6 – d) x2 + (0-C-2) x+7-c. In this last result the compound expressions a-d, b-d, 6-2-2 are regarded as the coefficients of x3, x2 and a respectively. E.tample 2. In the expression - aRx - 7a+ały +3 – 2.c – ab bracket together the powers of a so as to have the sign – before each bracket. The expression=-(aRx – a’y) – (7a + ab) – (2x – 3) - a? (x2 - y) - a (7+b) – (2x – 3) EXAMPLES VII. c. In the following expressions bracket the powers of a so that the signs before all the brackets shall be positive : 1. Ax4 + 6x2 +5+ 25x – 5x2 + 2x4 – 3x. 2. 36x2 – 7 – 2x + ab +- 5AXP + cx – 4.x2 – bx}. 3. 2–7.203 + 5ax2 – 2cx + 9ax3 + 7x - 3x2. 4. 2026 – 3abx + 4dx - 36x4 – a2.25 + x4. In the following expressions bracket the powers of x so that the signs before all the brackets shall be negative: 5. ax2 + 5x3 – a x4 – 26x3 – 3x2 – 6.x4. 6. 7.23 – 3c2x – abx5 + 5ax + 7.25 – abcx3. 7. ax2 + a223 — 6x2 – 5x2 – cx3. 8. 36 x4 – bx — ax4 - cx4 – 5c-x – 7.x4. Simplify the following expressions, and in each result regroup the terms according to powers of x: 9. ax3 – 2cx – [b.x2 – {cx – dx – (bx3 + 3cx2)} – (cx2 – bx)]. 10.5ax3 - 7(bx – cx2) - {66.24 - (3ax2 + 2ax) – 4cx2). 11. ax2-3{- ax +3bx - 4[7c23 - $ (ax - 62%)]). 12. x5 – 4bx4 12ax – 4 (36x4 – 9 6.2005 2 13. X{x – 6 – x(a - bx)}+ax – X{x — X (ax – b)}. ст 63. In certain cases of addition, multiplication, &c., of expressions which involve literal coefficients, the results may be more conveniently written by grouping the terms according to powers of some common letter. Example 1. Add together axi - 26x2 + 3, bx - 303 - ? and 23 - Ax2 + cx. = ax3 – cac3 + 2003. - ax– 2632 – 22 + 6x + cx+3 Example 2. Multiply ax2 – 26x + 3c by px - q. =ap.3 – 26px2 + 3cpu – aqx? + 2bqx – 3cq EXAMPLES VII. d. Add together the following expressions, and in each case arrange the result according to powers of x: 1. ax3 – 2cx, bx2 – CX3 and cx2 – X. 2. 22 – 2-1, ax? — 623, bx + 23. 3. a 23 – 5x, 2ax2 – 5ar), 2.03 – 6x2 - ax. 4. ax2 + bxc - C, qx – r-px?, x2+2x+3. 5. p.23 - 2x, 22:2 - px, 1-2, px?+qx3. Multiply together the following expressions, and in each case arrange the result according to powers of x : 6. ax2 + 6x +1 and cx+2. 7. cx2 – 2x+3, and ax – b. 8. ax2 – bx –c and px+q. 9, 2x2 – 3x – 1 and bx+c. 10. ax2 – 26x + 3c and x - -1. 11. p.x2 – 23 – 9 and ax – 3. 12. 202 + ax2 – bx — c and 23 - ax2 - 6x +c. 13. ax3 – 22 + 3x – 6 and ax3 + x2 + 3x + b. 14. xt - AX3 – 6x2 + cx + d and x4 + ax – bx? - CX + d. CHAPTER VIII. SIMPLE EQUATIONS. 64, An equation asserts that two expressions are equal, but we do not usually employ the word equation in so wide a sense. Thus the statement x+3+x=2x +3, which is always true whatever value x may have, is called an identical equation, or an identity. The sign of identity frequently used is =. The parts of an equation to the right and left of the sign of equality are called members or sides of the equation, and are distinguished as the right side and left side. 65. Certain equations are only true for particular values of the symbols employed. Thus 3x = 6 is only true when w=2, and is called an equation of condition, or more usually an equation. Consequently an identity is an equation which is always true whatever be the values of the symbols involved; whereas an equation (in the ordinary use of the word) is only true for particular values of the symbols. In the above example 3x=6, the value 2 is said to satisfy the equation. The object of the present chapter is to explain how to treat an equation of the simplest kind in order to discover the value which satisfies it. 66. The letter whose value it is required to find is called the unknown quantity. The process of finding its value is called solving the equation. The value so found is called the root or the solution of the equation. 67. The solution of equations, and the operations subsidiary to it, form an extremely important part of Mathematics. All sorts of mathematical problems consist in the indirect determination of some quantity by means of its relations to other quantities which are known, and these relations are all expressed by means of equations. The operation in general of solving a problem in Mathematics, other than a transformation, is first, to express the conditions of the problem by means of one or more equations, and secondly to solve these equations. For example, the problem which is expressed by the equation above given is the very simple question, "What is the number such that if multiplied by 3, the product is 6?” In the present chapter, it is the second of these two operations, the solution of an equation, that is considered. |