3. If a gasoline motor car runs 17 miles in 45 minutes, how far will it go in 5 hours at the same rate? expressed in the form ad- br, by the law of the means and extremes. Since the products ad and bx are equal the whole expression, ad = bx, is called an equation, which consists of two parts, the first member on the left side and the second member on the right side of the equality sign. Any quantity or group of quantities which is separated from others by the + or called a term of the equation. The equation has the following properties: sign is (a) The same quantity may be added to or subtracted from each member without destroying the equality. In the equation, 2b=4k, let 8 be added to each member. Thus, 26+ 8 = 4k + 8. Let 4 and k 2. Then substituting these values in the above equation, the equation becomes (2 × 4) + 8 = (4 × 2) + 8 b = = 8+8=8+8 It is seen that the quantities within the parentheses, which represent the values of the members of the original equation, are equal and the final result shows that the equality has not been destroyed by adding the same quantity to both members. It can be shown in a similar manner that the same quantity may be subtracted from both members of the equation without destroying the equality. (b) Each member of the equation may be multiplied or divided by the same quantity without destroying the equality. For example, divide by 3 each side of the equation 3x+6c9m + 15. Let x = 25, c 5, and m = 10. Substituting these values in the = be and hole two -on of n is ach let Let ove ent the It is seen that the equality has not been destroyed by the division. Similarly it may be shown that the same root of each member may be extracted, or that each member may be raised to the same power without destroying the equality. When a term has no sign before it, the plus sign is understood. 69. Transposition. (a) Suppose it is required to solve for the unknown value of x in the equation 4x+2=6+3x. In this equation the unknown quantity is found in both members. In any equation containing the unknown quantity in both members it is common to arrange the unknown values on the left side of the equation, and the known values on the right. After combining the terms divide both sides of the equation by the coefficient of the unknown quantity. The above operation may be accomplished by subtracting 3x and 2 from both members of the equation which results in the following: 4x 2-3x26+3x-3x-2 combining terms x +0=4+0 x = 4, Ans. Again, solve for x in the equation 8x+4x+6=2x+8 or x = .2 Ans. (b) The above form of solution may be simplified by the use of a method known as transposition. All terms containing the unknown quantity, which are found on the right side, are transposed to the left, at the same time changing their signs; likewise all known terms found on the left side are transposed to the right, at the same time changing signs. If the sign is +, change to -, and if -, change to +. Examples. 1. Solve the equation, 4x + 2 = 6 + 3x. SOLUTION: Bring over the 3x to the left-hand side of the equation and change its sign from + to; also transpose the 2 to the righthand side and change its sign to ; thus, 70. Negative Quantities. If an ordinary thermometer is consulted it will be found that the scale divisions have opposite them numbers which increase both upwards and downwards from 0. All values below the zero point are considered negative and all values above are considered positive; thus a temperature of 6 degrees means that the mercury reads 6 degrees below zero; similarly 6 degrees means that the reading is 6 degrees above zero. The boiling point of water on the Fahrenheit thermometer scale is 212 degrees. On the other hand, the temperature of liquid air, which is a very cold body, is -292 degrees Fahrenheit. Thus it is seen that the 0 point is merely a point of reference. In the same manner, the O of the numerical system is considered as a figure of no value and all numbers with the + sign have positive values and all numbers with the - sign have negative values. Suppose a man's money in the bank is 300 dollars and his debts amount to 350 dollars; here his money in the bank may be considered as positive and his debts as negative. Thus + 300-350 = −50` dollars. This 50 shows that the man is in debt 50 dollars, or has virtually 50 dollars less than nothing. Again when a ship sails a miles to the north of the equator, it is sailing in a positive direction as compared with a course to the south of the equator. For example, if a ship travels 200 miles north it is said to have traversed + 200 miles; on the other hand if it turns about and sails in the reverse direction 250 miles, it then will have a position 50 miles to the south of the starting point, i.e., its position will be -50 miles. Expressing this in equation form gives 200-250=−50. Examples. 1. What is the difference in longitude between two places where the longitudes are -80° and +30°? SOLUTION. Since one position is 80° from the starting point in one direction and the other 30° in the opposite direction from the starting point, the difference between the two places will be represented by the equation: 80+30=110° Ans. 2. A man has bills receivable to the amount of 500 dollars, and bills payable to the amount of 1,000 dollars; how much is he worth? SOLUTION. Having bills receivable to the amount of $500, this is what he is worth; but having bills payable to the amount of $1,000, this is what he owes. He is actually worth then: $500-$1000-$500 Ans. or he is in debt $500 PROBLEMS FOR PRACTICE 1. The temperature at 6 P. M. is +14° and during the evening it grows colder at the rate of 4° an hour. Required the temperature at 9 P. M., at 10 P. M., and at midnight. 2. What is the difference in latitude between two places where the latitudes are +86° and -14°? 71. Parenthesis. The parenthesis has already been shown to indicate that the terms enclosed are to be considered as one quantity. The following rules indicate the proper use of the paren thesis. Rules. (a) When a parenthesis is preceded by the sign the parenthesis may be removed without making any change in the expression within the parenthesis. (b) When a parenthesis is preceded by the sign the parenthesis may be removed if the sign of every term within the parenthesis be changed. (c) When a number or letter immediately precedes or follows the parenthesis, with no sign between, the multiplication sign is understood. Examples. 1. Remove the parenthesis from the expression: 8+ (2 + 6). In this problem, 2 is added to 6 before the parenthesis is removed, and the result is added to 8 to give the total sum of 16. It should be noted that the parenthesis has been removed without changing the signs within the parenthesis. 2. Remove the parenthesis from a + (b + c − d). a + (b + cd) = a + b + c − d According to the rule, since the quantity (64) is preceded by a minus sign the removal of the parenthesis makes it necessary to |