In like manner if 3 times a quantity denoted by a or 3 a be added to 5 times the same quantity or 5a, the sum will obviously be 8a. Again if 37. 10s., that is 37. + 10s. be added to 21. 5s., the sum is 5l. + 15s. EXERCISES. What is the sum of the areas of two squares, the side of each being 3 inches? 32+32= 2 × 32 = 18 square inches. What is the area of two equal squares, the side of each being denoted by a? a2 + a2 = 2 a2. What is the sum of 2 x2y+2xy2-3x y2-5x2 y - 3 x2y — x y2? What is the general rule for addition? To add quantities of the same kind or similar quantities. Add their coefficients, and retain the common letters. When they are not similar, place them after one another with the sign of addition between them. How do you proceed in subtraction? By a similar mode of reasoning the rule for subtraction may be easily obtained. Thus 51. diminished by 21. is 31. Also 5 a diminished by 2 a is 3 a. Again, 5l. 12s. diminished by 21. 4s. is 31. 8s. Also 5 a + 12b diminished by 2a + 4 b is 3 a + 8 b. The principle employed here is so obvious that every pupil will be able at once to recognise and employ it. There is however, another class in subtraction which is not so easily perceived, and which consists in subtracting a quantity having minus before it. Example: From 10 take a number which is 2 less than 6. If 6 be taken from 10 the remainder may be expressed by 10 – 6. Hence the true remainder may be expressed by 10−6+ 2=6. If we represent the quantities by letters, then b c taken from a will be expressed by a- b+c. The principle here employed may be thus expressed. If the quantity in the subtrahend have minus before it, it will have plus before it in the remainder. From 5a+3x-2 b take 2 c-4y. The quantity to be subtracted with its signs changed, is — 2 c +4y: therefore the remainder is 5 a + 3 x 26-2c+4 y. What is the practical rule for subtracting one algebraic expression from another? Change the signs of all the terms in the subtrahend from plus to minus and from minus to plus, and then simplify the result. From 4 a2-3 b + c Take a2-2b <-c The remainder is expressed by 4a2-3b+c - a2 +2b+c, which when simplified becomes 3 a2 — b + 2 c. The mode of simplifying is obvious. Thus 4 a2 a2 is 3 a2; 2b to be added and 3b to be subtracted is the same as one b or simply b to be subtracted, and c to be added, is 2 c to be added. How is algebraical multiplication performed? Multiplication being nothing more than the addition of equal quantities, the rule for this process will be easily obtained. Thus a + a + a + a is 4 times a or 4 a. Also 3 a + 3 a is 2 times 3 a 3 a x 26 a. Again, 4 × 4 is expressed by 4a and a × a by a2. Therefore 2 a× 3 a is 6 a. Also 2a xa x 3 a × a × a or 2 a2 × 3 a3 is 6 αδ. Multiply 3 ab- 2 ac+d. by 4 a Product 12 a2 b 8 a2c + 4 a d. If the numbers be expressed by different letters the product of the coefficients will obviously be placed before the letters written in succession. Thus 2 a 3b is 6 a b ; 3 a2 × 2 a b × 4 c = 24 a3 b c. to add ( a b) to Product a2+2ab + b2 d); now this is To multiply (a - b) by (cd) is itself as often as there are units in (c evidently done by adding it c times, and subtracting it d times. But (ab) added c times is = and (a — b) subtracted d times is = · a—b) × (c–dis = ac - bc ac be -- ad + bd ad + bd whence -a {+x+=+ac} i. e. like signs produce +. c and {0}i.e. unlike signs produce – What is the rule for performing division of algebra ? The operation of division being the reverse of that of multiplication, the rule is as follows: To the quotient of the coefficients annex the quotient of the letters, and the complete quotient will be exhibited. divided by 8; also a a and a a a may both be divided by aa; bb and b, may both be divided by c. Hence the fraction is simplified or reduced to b a c The general rule is therefore to divide both numerator and denominator by every quantity which will ivide both without leaving a remainder. SECT. II. On the solution of Questions by means of Simple Equations. What is meant by an equation? An equation is merely an expression of equality between two quantities, and is expressed by placing the sign between them. Thus 48 12 is an equation; also 3 x + 5 = 26 is an equation expressing the equality of the quantities 3 x+ 5 and 26. Given the equation 6 x 3=2x+5. Transposing the 2x and changing its sign, we have the equation 6 x 2 x 35. Transposing the known term 3, changing its sign, we have finally 6 x -2x= 53. Let x+8=15: subtract 8 from each side of the equation, and it becomes x+8 -8 15 - 8: but 8 8 = 0..x = 15 87. Let x-7= 20: add 7 to each side of the equation, then x — -7+7= =20+ 7: but 7+7=0..x = 20 +7= 27. Let 3 x.5= = 2 x + 9; add 5 to each side of the equation, and it becomes 3 x 5 + 5 = 2 x + 9 +5, or 3 x = 2 x + 9 +5. Subtract 2 x from each side of this latter equation, then 3 x 2x=2x- 2x+9 +5: but 2 x 2x = 0..3 x -2x=9+5. Now 3 x 2x= = x, and 9514, hence x = 14. What number is that to which if 5 be added the sum will be 9? Let x= the number required. Then x + 5 = 9, by the question. Now since x is less than 9, by the number 5, it follows that x = 9-5 = 14, the number required. What number is that from which if 5 be taken the remainder will be 4? Let x denote the number required. Then x - 5 = 4, by the question. Now since x is greater than 4 by the number 5, it follows that x=4+5=9, the number required. From these examples it is obvious if a number have plus before it on one side of an equation it may be carried to the other by placing minus before it, and the contrary. What number is that which being multiplied by 4, the product will be 12? Let x represent the number required, then 4x 12 by the question. Now since it requires 4 times x to make 12, x will be the fourth part of 12, or x == =3, the number required. 12 4 The learner will perceive from this example that if the unknown number be multiplied by a known one, it may be freed from it, by dividing the other term by that number. |