4. The number of pupils in a class is n. Five pupils are withdrawn. Express as a binomial the number left. 5. A man had a acres of land and sold s acres. How many had he left? 6. The difference of two numbers is 20. The smaller number is n. What is the larger? 7. Express in symbols: & of a number n diminished by 4. 8. Five times a number x is increased by 3, and the sum is divided by 7. Write the result in symbols. Write the following polynomials in the simplest form. 9. 2.x +78 -4.x+6x. Solution: 2x+7x - 4x+6x=2x+7x+6x - 4x = 15x - 4x = 11x. 10. 3a+7a+2a-a. 11. jm+&m+fm-im. 12. 2.3t+1.81—.9t+6.1t. 13. 20m+6m+17n3n. Solution: (20m+6m)+(17n-311) = 26m+14n. 14. 14x — 3.6x+8.4y - 1.9y. 15. 1ğa-1a+36-1b. 16. 12.6c-.25c+18.3c. 17. (8x+4x)+(10.x - 2.)+3. 18. ja+ka+ta+5. Find the value of each of the following for a=6, b=3, c = 2. 19. a+26+3c. 3c+2b-a+1 20. 2a+ 23. 8c-b-a 2 21. a-c+26 3+2a 24. la-b+c+6 a-36+ c 6.2a-2.4b+c Qa+1.25+.60 22. 20+1.2a- .66 a+b+c 25. EQUATIONS 45. How to solve equations. We have seen ($$30– 32) that numerical facts may be represented by equations. For example, the equation c=30n states that the number of cents paid for oranges is 30 times the number of oranges. In studying perimeters ($34) we were able to find the lengths of sides of polygons by means of equations, such as 6x=120. In the study of mathematics the equation is a powerful instrument in solving problems and in representing numerical facts in a brief form. Hence, we must learn to work with equations. An equation, as 6x=30, states the equality of the numbers 6x and 30. The number 6x to the left of the equality sign is called the left side, or the left member, of the equation; the number to the right is the right side, or right member. The literal number, x, is called the unknown number. The process of finding a value of the unknown number for which both members are the same is called solving the equation. EXERCISES Solve the following equations: 1. 2x=19. Solution: 2x=19. Dividing each member by 2, we have .: X=9.5. 2. 3x = 18. 4. 5x = 20. 6. 2.5x = 50. 3. 2x=11. 5. 7x=13. 7. 3.75x = 100. For each of the following problems first state the equation and then solve it. Arrange your work as in the solution of Exercise 8, below. 8. A certain number multiplied by 6 gives the product 42. Find the number. Solution: Let n be the required number. Then the problem is briefly stated in the form of the equation on=42. To find n, divide each member of the equation by 6. en 42 This gives 6 ..n=7. 9. A train traveling at the rate of 35 miles an hour made a distance of 75 miles. How much time did it take? 10. A field containing ã of an acre is sold for $300. What is the price per acre? 11. In a given time the minute hand of a clock passes over 12 times as many minute spaces as the hour hand. Over how many minute spaces does the hour hand pass in 50 minutes? the rate of 18 12. How long will it take a man to ride 60 mi. mi. an hour? 13. A man earns $125 in 16 days. How much does he get per day? 46. Solving equations by using an axiom. In the solution of equations ($45) it has been taken for granted that when both members of an equation are divided by the same number, the quotients are also equal. it is assumed that 2.0 10 2 2 or that x=5. This assumption may be stated in the form of a general principle as follows: If equal numbers are divided by the same or equal numbers, the quotients are equal. Such statements when assumed to be true are called arioms. The equation above is solved by dividing both members by 2. The principle used in the solution of the equation is called the division axiom. This axiom is to be used in solving each of the equations below. EXERCISES Solve the following equations explaining each step, and arrange the work as shown in Exercise 1, below: Solution: Authorities: 1. 2a=16 2α 16 If equal numbers are divided by the same number the quotients are equal. In the following equations find the values of the unknown numbers approximately to the nearest third figure: 8. 1.23y=532. 12. .57r=24.2. 9. 287x = 5.89. 13. 1.32t=226. 10. 21.2n=62.1. 14. 1188=237. 11. 7.5k=28.2. 15. .231x =462. Solve the following problems by means of equations, arranging the work as shown in Exercise 16, below: 16. John is able to solve twice as many problems as James. Mary can solve three times as many as James. Together they solve 48 problems. How many problems does each solve? Solution: Let x be the smallest number, i.e., the number of problems James solves and 3x is the number Mary solves 6x =48, by combining similar terms Notice that the solution of the problem involves the following steps: a. The problem is read carefully, to find what number the problem calls for. In this case there are three unknown numbers. b. One of the unknown numbers is denoted by a letter, as "x." In this case x denotes the number of problems solved by James. C. The other unknown numbers are now expressed in terms of "x.” Thus, John and Mary solve 2x and 3x, respectively. d. Then the equation is formed and solved. These suggestions, if followed, will be helpful in solving the problems below: 17. A sum of $32 is to be broken up into two parts, one part being seven times as large as the other. Find the two parts. 18. Sixty rods of fence are available to inclose a rectangular field. The field is to be five times as long as it is wide. Make a sketch of the field and find the dimensions. 19. A man is three times as old as his son. The sum of their ages is 48 years. Find the age of each. |