CASE II. When the Quantities are Like, but have Unlike Signs ; ADD the affirmative co-efficients into one sum, and all the negative ones into another, when there are several of a kind. Then subtract the less sum, or the less co-efficient, from the greater, and to the remainder prefix the sign of the greater, and subjoin the common quantity or letters. CASE III. When the Quantities are Unlike. HAVING collected together all the like quantities, as in the two foregoing cases, set down those that are unlike, one after another, with their proper signs. Add a+b and 3a-5b together. Add 5a-8 and 3a-4x together. Add 6x-5b+a+8 to -5a-4x+4b-3. Add a+26-3c-10 to 3b-4a+5c+10 and 5b-c. Add a+b and a-b together. Add 3a+b-10 to c-d-a and -4c+2a-36-7. Add 3a2+b2-c to 2ab-3a2+bc-b. Add a3+b2c-b2 to ab2-abc+b2. Add 9a-8b+10x-6d-ic+ 50 to 2x-3a-5c+4b+6d -10. SUBTRAC This rule is founded on the consideration, that addition and subtraction are opposite to each other in their nature and operation, as are the signs + and -, by which they are expressed and represented. So that, since to unite a negative quantity with a positive one of the same kind, has the effect of diminishing it, or subducting an equal positive one from it, therefore to subtract a positive (which is the opposite of uniting or adding) is to add the equal negative quantity. In like manner, to subtract a negative quantity, is the same in effect as to add or unite an equal positive one. So that, by changing the sign of a quantity from + to-, or from to +, changes its nature from a subductive quantity to an additive one; and any quantity is in effect subtracted, by barely changing its sign. bxy-30 7x3-2(a+b) 7xy-50 2x2-4(a+b) 3xy3 +20a/(xy+10) From a+b, take a-b. From 4a+4b, take b+a. From 8a-12x, take 4a-3x. From 2x-4a-2b+5, take 8-5b+a+6x. MULTIPLICATION. This consists of several cases, according as the factors are simple or compound quantities. CASE I. When both the Factors are Simple Quantities; FIRST multiply the co-efficients of the two terms together, then to the product annex all the letters in those terms, which will give the whole product required. Note. Like signs, in the factors, produce + and unlike signs, in the products. EXAMPLES. That this rule for the signs is true, may be thus shown. 1. When a is to be multiplied by +c; the meaning is, that + a is to be taken as many times as there are units in c; and since the sum of any number of positive terms is positive, it follows that + a + c makes + ac. 2. When When one of the Factors is a Compound Quantity. MULTIPLY every term of the multiplicand, or compound quantity, separately, by the multiplier, as in the former case; placing the products one after another, with the proper signs; and the result will be the whole product required. 2. When two quantities are to be multiplied together, the result will be exactly the same, in whatever order they are placed; for a times c is the same as c times a, and therefore, when a is to be multiplied by + c, or + c by — a, this is the same thing as taking a as many times as there are units in + c; and as the sum of any number of negative terms is negative, it follows that - ax + e, ora Xc make or produce ac. 3. When a is to be multiplied by -e: herea is to be subtracted as often as there are units in e: but subtracting negatives is the same thing as adding affirmatives, by the demonstration of the rule for subtraction; consequently the product is c times a, or + ac. Otherwise. Since a a= =0, therefore (a — a) X — c is also = 0, because O multiplied by any quantity, is still but 0; and since the first term of the product, or a Xcis=- ac, by the second case; therefore the last term of the product, or - a X c, must be ac, to make the sum = ·0, or acac =0; that is, a X -C= + ac. EXAMPLES. |