20. A certain capital earns each year a dollars interest. capital were diminished by m dollars, it would earn only interest. Find the capital and the percentage. If this dollars $1.25 of the first kind and $2.00 21. A has two kinds of silver. of the second kind melted together give 134 parts of pure silver; $.75 of the first and $2.50 of the second kind melted together give 13 parts of pure silver. How fine was each kind of silver? 71 22. A man has two quantities of gold of different purity. pounds of the first kind and 190 pounds of the second kind give 800 parts of pure gold; 171 pounds of the first kind and 95 pounds of the second kind give 900 parts of pure gold. How pure is each kind of gold? 23. A miller expended $375 for wheat and rye. He paid $1.875 for a bushel of wheat, and $1.25 for a bushel of rye. Had he sold 4 weeks earlier, he would have gained $15.25, because, at that time, wheat was 2 cents and rye 10 cents higher per bushel than now. How many bushels were there of wheat and rye respectively? 24. What value do a dollar and a rouble have in marks, and in what ratio does the dollar stand to the rouble, if 48 roubles plus three marks are equal to 36 dollars, and 1 dollar and 1 rouble are exactly 7 marks? 25. The sum of two numbers is 15390, the first of which contains one figure, the second, five figures. If the first is placed to the left before the second, the number thus formed is 4 times as large as the number which is formed by placing the first to the right, behind the second. What are the numbers? 26. Two numbers have a given product. If the first were 8 less and the second 25 larger, their product would be increased by 5000. If the first were 12 greater and the second 25 less, their product would be 4000 less. What are the two numbers? 27. A farmer brought eggs to market, and hoped to sell them at a certain price. Had he sold the eggs for cent each more than he had hoped, then he would have realized his total price, if 12 of the eggs had been broken on the way. But if he had been obliged to sell the eggs at cent each cheaper than he had thought, then he would have needed 12 eggs more, in order to receive 3 cents more than he had at first hoped. How many eggs did he have, and what should each egg sell for? 28. The difference of the squares of two numbers is 840. If each number were 3 larger, the difference of their squares would be 900. What are the numbers? 29. Two amounts of money, one of which is $1000 more than the other, are lent at different rates-the second at % higher than the first-and both amounts earn the same amount of interest. If the first amount were lent at the rate at which the second was lent, and the second at the rate of the first, then the first would earn $95 more than the second. How large were the two capitals, and at what rate was each lent? 30. A composition of lead and zinc, which weighs 149 pounds, loses 18 pounds in water. How many pounds are there in each metal, if a quantity of 11 pounds of lead and one of 63 pounds of zinc each loses 1 pound in water? 31. A composition of two metals loses p pounds in water. How many pounds are there of each metal, if a pounds of the first loses m pounds in water, b pounds of the second loses pounds in water, and the entire mass weighs q pounds? 32. There are two numbers, one of which contains two figures and the other four. If the second is divided by the first, the quotient is 204, with a remainder of 1; if a number is formed by writing the first before, and to the left of, the second, this number is half as large as the number which is found by writing the second number before, and to the left of, the first. What are the numbers? 33. A boy made a cork belt in order that he might swim with greater ease. The boy and the belt weigh 139 pounds, and the boy is of exactly such weight that he can keep his head, which weighs 12 pounds, out of water, without its being raised above or lowered into the water more than is necessary for the movement of his arms and feet in swimming. How much did the boy and the cork belt each weigh, if 120 pounds of the boy's body immersed in the water weighed 3 pounds and the specific weight of the cork belt is 0.24? 34. The fore wheel of a carriage makes six (a) revolutions more than the hind wheel in going 120 (b) yards; if the circumference of the fore wheel be increased by (1) of its present size, and the circumference of the hind wheel by () of its present size, the six (a) revolutions will be changed to four (c). Find the circumference of each wheel in both cases. CHAPTER VII GENERAL SOLUTION OF A SYSTEM OF TWO EQUATIONS IN TWO UNKNOWN QUANTITY HAVE A COMMON ROOT 226. In 203, 205 (1), reference was made to the forms a 0 and which may occur in the solution of an equation of the first degree. The meaning of the forms when they occur in the solution of simultaneous equations of the first degree is here treated. review the results already obtained. First 227. Any equation of the first degree in one unknown quantity can be reduced to the form ax=b, from which x = b. α When a ap proaches 0, x becomes a quantity which may be as large as is desired. In this case we have 0xb, an equation which can not be satisfied by any finite value of x because, as long as x is finite, x0 =0. But the equation requires that 0.x shall be finite and equal to b. Therefore a solution is impossible. Again, if a = b = 0, then ax = b takes the form 0x = 0, which is indeterminate, since for any finite value of x, Ox 0. Therefore an infinite number of values of x would satisfy the equation. In case b =0, and a = 0, then x = 0 is a possible solution of the equation. GENERAL SOLUTION OF A SYSTEM OF Two EQUATIONS IN Two UNKNOWN QUANTITIES 228. Two equations in two unknown quantities can always be put in the form, System I is, therefore, equivalent to system II, and the second equation involves but one unknown quantity. Therefore the solution and the discussion of a system of two equations of the first degree in two unknown quantities resolves itself into the solution and the discussion of an equation of the first degree in one unknown quantity. Suppose ab'-a'b=0. Equation (4) has the root THE COMPOSITION OF THE FORMULAE 229. The composition of these formulae is easy to exhibit. The values of x and y have the same denominator, ab' — a'b. The denominator is formed by taking the product of the coefficients of the unknown quantities crosswise, first from left to right and thenfrom right to left, and then taking the difference of the products. The numerator of x is formed by substituting, in ab'-a'b, c and c' for a and a', the coefficients of x in the two given equations. The numerator of y is formed by substituting in ab'— a'b, c and c' for b and b', the coefficients of y in the two given equations, a is changed into a', and a' into a, b into b', and b 'into b, c into c', and c' into c, the first equation will be transformed into the second and the second into the first, and the system will not be changed. Hence, if the same changes are made in formulae III, the values found for x and y should be the same. This is exactly what happens. The expressions, Hence the values of x and y are not changed. 231. Suppose that in the given system y and y into x, a into b and b into a, a' into b' and formed a second system of equations, c is changed into b' into a'; there is which does not differ from the first except in the order of the terms in the first members of the equations. The value of x found from system I was b'c be' x= ab' a'b |