Divide both members by -3, and x=4. 4. Given 5x+22-2x=31, to find x. Dividing both members by (a+b), gives x=m. 9. Given ax+dx=a--c, to find x. ac Ans. x= 10. Given 3(x+1)+4(x+2)=6(x+3), to find x. a+d In the first place, drop 16 from both members, according to the general rule. Then 3x x X 4 = 2 + 8 + 1 Multiply both members by 8, the least common multiple of the denominators, and we have +2=3, to find a. Ans. x 12. 18. Given 1x+x+1x=39, to find the value of x. Here are no scattering terms to collect, and clearing of fractions is the first operation. By examination of the denominators, 12 is obviously their least common multiple, therefore, multiply by 12. 19. Given 1x+3x+1x=a, to find x. This example is essentially the same as the last. It is identical if we suppose a=39. Now if a be any multiple of 13, the problem is easy and brief in numerals. 20. Given 1-5+1x+8+}x—10—100—6—7 to find the value of x. Collecting and uniting the numeral quantities, we have Multiply every term by 60, and we have 20x+15x+12x=94.60 24. Given y+y+y=82, to find y. 25. Given 5x+x+x=34, to find x. . N. B. In solving 21, 22, 23, 24 and 25, take 19 for a model, and write a to represent the second members of the equations, to save numeral multiplications. Ans. x 120. Ans. y= 84. Ans. x= 6. (ART. 54.) When a minus sign stands before a compound quantity, it indicates that the whole is to be subtracted; but we subtract by changing signs, (Art. 5). The minus sign in the last example, does not indicate that the a before x-1 2 is minus, but that this term must be subtracted. When the term is multiplied by 4, the numerator becomes 2x-2, and subtracting it, we have -2x+2. PROPORTION. (ART. 55.) Sometimes an equation may arise, or a problem must be solved through the aid of proportion. Proportion is nothing more than an assumption that the same relation, or the same ratio exists between two quantities as exists between two other quantities. Quantities can only be compared when they are alike in kind, and one of them must be the unit of measure for the other. Thus, if we compare A and B, we find how many times, or part of a time, A is contained in B, by dividing B by A, thus, B A=", or B=rA That is, a certain number of times A is equal to B. Now if we have two other quantities, C and D, having the same relation or ratio as A to B, that is, if D=rC, Then A is to B as C is to D. But in place of writing the words between the letters, we write the signs that indicates them. But in place of B and D, write their values rA, and rC. Multiply the extreme terms, and we have rCA. Multiply the mean terms, and we have rAC. Obviously the same product, whatever quantities may be represented by either A, or r, or C. Hence, to convert a proportion into an equation, we have the following RULE. Place the product of the extremes equal to the product of the means. (ART. 56.) The relation between two quantities is not changed by multiplying or dividing both of them by the same quantity. Thus, ab: 2a: 26, or more generally, a:b::na: nb, for the product of the extremes is obviously equal to the product of the means. That is, a is to b as any number of times a is to the same number of times b. We shall take up proportion again, but Articles 55 and 56 are sufficient for our present purpose. EXAMPLES. 1. If 3 pounds of coffee cost 25 cents, what will a bag of 60 pounds cost? Ans. 500 cents. Ans. It will cost a certain number of cents, which I designate by x, and the numerical value of x can be deduced from the following proportion: Pounds compare with pounds, as cents compare with cents. That is, these different kinds of quantities must have the same numerical ratio. Without the x, this is the rule of three in Arithmetic, because there are three terms given to find the fourth; and in Algebra we designate the fourth term by a symbol before we know its numerical value, which makes the proportion complete. By the rule (Art. 55), 3x=60.35 Hence, when the first three terms of a proportion are given to find the fourth, multiply the second and third together, and divide by the first. In Arithmetic it requires more care to state a question than it does in Algebra, because in the former science we have not so much capital at command as in the latter. In Algebra it is immaterial what position the unknown term 664140 |