30. Find the value of x3 – 8y3 +29z3 +18xyz when 2y=x+3z, and z=5. 31. Solve the equations: 2x+3y= 5.1 10x - 6y=11. Problems leading to simultaneous equations. We have already found that to solve a simultaneous equation it is essential to have as many independent equations as there are unknown quantities to determine. Therefore in the solution of any problem which produces a simultaneous equation it is necessary in the statement of the question that there should be as many equations involved as there are unknowns to be determined. Ex. 1. If 3 be added to the numerator of a certain fraction, its value will be, and if 1 be subtracted from the denominator its value will be. What is the fraction? Let x be the numerator, and y the denominator of the fraction. Add 3 to the numerator, then x+3 1 y 3 3 1. Find the fraction which becomes equal to if its denominator be decreased by 6, and becomes equal to if its numerator and denominator be increased by 124. 2. A bill of £13. 12s. 6d. was paid with 40 coins, some of them half-crowns and the others half-sovereigns: find the number used of each kind. 3. If 10 yards of silk and 7 yards of satin cost £5. 6s. 4d.; and if 3 yards of the satin cost as much as 4 yards of the silk, find the price of a yard of each. 4. A certain number, consisting of two digits, exceeds four times the sum of its digits by 3; if the number be increased by 18, the result is the same as if the number formed by reversing the digits were diminished by 18; what is the number? 5. Find the fraction which is equal to when 10 is added to its numerator, and which is equal to when 4 is subtracted from its denominator. 6. The numerator of a certain fraction is 4 less than the denominator; if 10 be subtracted from the numerator, or if 30 be added to the denominator, the resulting fractions would be equal: find the original fraction. 7. (i) The expression ax2 + bx - 30 is equal to 240 when x=5, and is equal to 100 when x=-2; find the values of a and b. (ii) If Q varies as H3, and Q is 7.26 when H is 1·5, find H when Q is 5.68. 8. A quantity x equals the sum of two quantities, one of which varies as t the other as t2; when t equals 3, x equals 159, and when t equals 5, x equals 425; find the value of x when t equals 10. 4. A contribution of £125 was raised by A, B, and C; A gave twice as much as B, and B gave £15 more than C; what did each give? 5. A man bought some tea of two sorts, 3 lbs. of the first costing as much as 4 lbs. of the second, while 9 lbs. of the first and 8 lbs. of the second together cost £2. 5s.: what was the price of each per lb.? Solve the equations : 5 – 4 13. What values of x will make the expression x2+4.8x+2.87 have the value zero? 14. Divide £490 among A, B, and C, so that B shall have £2 more than A, and C as many times B's share as there are shillings in A's share. 15. In a certain proper fraction the difference between the numerator and the denominator is 8, and if each be diminished by 7, the fraction becomes equal to . Find the fraction. 16. An army in a defeat loses one-sixth of its number in killed and wounded, and 4,000 prisoners. It is reinforced by 3,000 men, but retreats losing a fourth of its number in doing so. remains 18,000 men. What was the original force? There 17. The expression ax2+ bx -- 30 is equal to 240, when x equals 5, and is equal to 100, when x equals -2; find the values of a and b. 18. The electrical resistance of a wire of given material varies directly as the length and inversely as the square of the diameter. Find the length and diameter of a wire which is to have threefourths of the electrical resistance, and one-third of the weight of a wire of the same material 50 ft. long and 0.03 in. diameter. Summary. Identity. An equality involving only an algebraic operation between two quantities is called an identity. Simple Equations.-A statement of equality between two algebraic expressions is called an equation, and is called a simple equation when it involves only the first power of the unknown quantity. In an equation the results are still equal if corresponding operations (addition, subtraction, multiplication, and division) are performed on both sides of the equation. Simultaneous Equations. In an equation which contains two unknown quantities x and y, an indefinite number of pairs of values may be found which will satisfy the equation; but if a second equation be given, only one pair can be found which will simultaneously satisfy both equations. CHAPTER IX. GRAPHIC METHODS. SCALES AND THEIR USE. Graphic methods. In the calculations which have to be made continually by the practical man, there is always the risk, even when great care is exercised, of errors finding their way into the work. Hence some trustworthy and independent method by which the calculations can be checked is desirable. Such a check is furnished by graphic methods, in which drawing instruments can be used for all, or nearly all, the problems with which the practical man has to deal. With care and the use of good scales, results are easily and readily obtained with considerable accuracy. Scales and their use. The majority of the problems considered are supposed to be solved by the process known as drawing to scale. In making a drawing of any large object, such as a building, it would be inconvenient, if not impossible, to make it as large as the object itself. In other words, to draw it full size is impossible, but if a drawing were made in which every foot length of the building were represented on the drawing by a length of half an inch, the drawing would be said to be drawn to a scale of inch to a foot, or to a scale of 4. By means of a suitable scale any required dimension could be obtained as readily as if the drawing were made full size. In a similar manner, if the drawing were made so that every length of 3 inches on the drawing represented an actual length of 12 inches, the scale would be said to be. This fraction of, or , etc., is called the representative fraction of the scale. Hence, Representative fraction of a scale number of units in any line on the drawing number of units the line represents |