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Elementary Algebra and Geometry for Schools

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HOME : Mathematics - Geometry

Root of an equation. In the preceding exercises we found the value of the unknown letter. We then "checked" our answer by substituting it for the unknown letter in the original equation in order to determine whether it satisfied the original equation.

  • A value of the unknown letter which satisfies the equation is called a root of that equation.

Naming the unknown. In ancient Egyptian algebras, the unknown quantity in the equation was referred to as "the heap." Other ancient writers called the unknown quantity "the thing." AI-Khowarizmi, about 830, called it "the thing" or "the root" (of a plant). For a long time mathematicians used both terms "the thing" (res) and "the root" (radix), but finally the word root became the accepted name for the unknown quantity in an equation.

ILLUSTRATIVE EXAMPLE: Solving an equation by collecting like terms Solve the equation 2x + 3x = 24 - 4.

EXERCISES: Solving equations

Solve each of the following equations and check your answer:

Answers : 1. 2. 2. 6. 3. 8. 4. 9. 5. 3. 6. 9. 7. 2. 8. 1. 9. 3. 10. 2. 11. 9. 12. 3. 13. 2. 14. 5. 15. 10. 16. 5. 17. 3. 18. 19. 19. 1.5. 20. .01.

REVIEW: EXERCISES

1. A certain number is represented by x. How would you represent a number that is twice x?

2. A certain number is represented by y. How would you represent a number that is 4 more than twice y?

3. A certain number is represented by z. How would you represent a number that is 2 less than three times z?

4. If x represents a certain number, how would you represent a number that is 5 more than twice x?

5. One number is five times a second number. If x represents the second number, how would you represent the first number?

6. One number is 7 more than three times another number. If y represents the smaller number, how would you represent the larger number?

7. A man has 4 more nickels than quarters. If the number of quarters that he has is represented by x, how would you represent the number of nickels that he has?

8. A man has three times as many dimes as pennies. If the number of pennies that he has is represented by y, how would you represent the number of dimes that he has?

9. A man has 2 more nickels than dimes and twice as many quarters as dimes. If the number of dimes that he has is represented by x, how would you represent the number of nickels that he has? How would you represent the number of quarters that he has?

10. John is 3 years older than James. If the number of years in James's age is represented by y, how would you represent the number of years in John's age?

11. Mary is twice as old as Joan. If the number of years in Joan's age is represented by x, how would you represent the number of years in Mary's age?

12. Mrs. Jones's age is 4 years less than twice her daughter's age. If the number of years in the daughter's age is represented by x, how would you represent the number of years in Mrs. Jones's age?

Answers : 1. 2x. 2. 4 + 2y. 3. 3z -2. 4. 2x + 5 . 5. 5x. 6. 7 + 3y. 7. 4 + X. 8. 3y. 9. 2 + x; 2x. 10. y + 3. 11. 2x. 12. 2x -4.

REVIEW: Completion test

On another sheet of paper write the numbers of the incomplete statements in a column. After each number write the algebraic expressions which correctly complete the statement.

1. One number is twice another. One of these numbers can be represented as --and the other number as--.

2. One number is 10 larger than another. The smaller number can be represented as --and the larger number as--.

3. One number is 5 less than another number. The larger number can be represented as --and the smaller number as--.

4. One number is six times another number. The smaller number can be represented as --and the larger number as--.

5. One number is half of another number. The larger number can be represented as --and the smaller number as--.

6. One number is 8.2 times as large as another number. The smaller number can be represented as --and the larger as --.

7. One number is 8 larger than another number. The smaller number can be represented as --and the larger number as--.

8. One number exceeds another number by 2. The smaller number can be represented as--and the larger number as--.

9. One number is 6 more than twice another number. The smaller number can be represented as --and the larger number as --.

10. One number is five less than three times another number. One of the numbers can be represented as --and the other as--.

Answers : 1. x,2x. 2. x, x + 10. 3. x, x -5. 4. x, 6x. 5. x, x/2 6. x, 8.2x. 7. x, x + 8. 8. x, x + 2. 9. x, 6 + 2x. 10. x, 3x -5.

Solving problems. Many problems can be solved with equal facility by using either arithmetic or algebra; many other problems are exceedingly difficult by arithmetic, but are quite easy by algebra. Most of the problems that you will solve in the next few exercises can be done easily by either arithmetic or by algebra, but you must learn to solve them by algebra so that you will be ready to solve problems that are difficult, if not impossible, to solve by arithmetic.

"Word" problems. These are problems that are written in words rather than in algebraic symbols. When you are given a relationship between two llumbers, you can now represent the two numbers in terms of a single letter. For example, when you are given the fact that one number is twice another, you represent one of the numbers as x and the other number as 2x.

The first step in solving "word" problems is to represent the unknown quantities, or numbers, in terms of a single letter. The second step in solving problems that are stated in words is to translate a verbal statement into an algebraic equation.

Finally, the third step in solving word problems is to solve the equation that you have written, and thus be able to answer the question of the problem.

ILLUSTRATIVE EXAMPLE: Solving word problems

Twice a certain number, when increased by 4, is equal to 56. Find the number.

Solution. First we represent the certain number by x; then twice the number can be represented by 2x. Next we translate the words of the statement into an equation:

Subtracting 4 from both members 2 x + 4 - 4 = 56 - 4

Collecting like terms 2x = 52

Dividing both members by 2     2x/2 = 52/2

Simplifying x = 26

Answering the original question, the certain number is 26.

Check. Twice the certain number is twice 26, or 52; twice the certain number when increased by 4 is 52 increased by 4, or 56; thus the original statement checks.

EXERCISES: Problems

1. If a certain number is increased by 23, the result is 42. Find the number.

2. Three times a number is equal to 66. Find the number.

3. If 18 is subtracted from a number, the result is 14. Find the number.

4. The sum of five times a certain number and three times that number is 72. Find the number.

5. Twice a certain number, increased by 13, is equal to 25. Find the number.

6. If three times a certain number is increased by 11, the sum is 53. Find the number.

7. If five times a number is diminished by 31, the result is 4. Find the number.

8. If 18 is added to four times a number, the result is 70. Find the number.

9. One number is six times as large as another. If the smaller number is subtracted from the larger, the result is 35. Find the two numbers.

10. One number is three times as large as another number. If the sum of the two numbers is 216, what are the numbers?

11. One number is four times another number. If the difference of the two numbers is 108, find the numbers.

12. One number is seven times another number. The larger number increased by 18 is equal to 116. Find the smaller number.

13. One number is twice another number. The larger number decreased by 25 is equal to 121. Find the smaller number.

14. The larger of two numbers is 7 less than three times the smaller number. If the larger number is 47, find the smaller.

15. One number is 2.2 times as large as another number. If the sum of the two numbers is 48, what are the numbers?

Answers : 1. 19. 2. 22. 3. 32. 4. 9. 5. 6. 6. 14. 7. 7. 8. 13. 9. 7, 42. 10. 54, 162. 11. 144, 36. 12. 14. 13. 73. 14. 18. 15. 15, 33

A letter always represents a number. In your previous study, and in your work in equations in this page, you have learned to represent numbers by letters. It is important now to stress the fact that these letters you have been using always represent numbers. A literal number can represent the number of pounds of a weight, just as the number 5 can represent the number of pounds of potatoes; a literal number can represent the number of years in a man's age, just as the number 14 can represent the number of years in a boy's age; a literal number can represent the number of feet in the length of a room, just as the number 300 can represent the number of feet in the length of a football field. In the word problems that you are going to do next, you will use x (or some other letter) to represent a number of coins, or a number of dollars, or a number of years. Always remember that a letter represents a number. It is better to say that x represents the "number of nickels" or the "number of miles" than it is to say that x represents "the nickels" or "the distance."

ILLUSTRATIVE EXAMPLE: Word problems

A father is three times as old as his son, and the sum of their ages is 64 years. Find the age of each.

Solution. Let x represent the number of years in the son's age; then the number of years in the father's age can be represented as 3x. Since the sum of their ages is 64 years, we can write the equation:

x + 3x = 64

Collecting like terms 4x = 64

Dividing both members by 4 x = 16

Answering the questions, the number of years in the son's age, x, is 16; hence the son is 16 years old.

The father's age, 3x, is 48; hence the father is 48 years old.

Check. Does 16 +48 equal 64?

EXERCISES: Problems

1. A man has some coins consisting of nickels and dimes, and he has twice as many nickels as he has dimes. Find the number of each kind of coin that he has if he has 36 coins in all.

2. A father is three times as old as his daughter. If the sum of their ages is 52 years, how old is each?

3. A theatre sold twice as many children's tickets as adults' tickets. If the total number of tickets sold was 411, how many adults' tickets were sold?

4. Bill is twice as old as his sister Betty, and the sum of their ages is 21 years. Find the age of each.

5. A basketball team won three times as many games as it lost. If the team played 24 games, how many did it win?

6. By driving throughout a morning and an afternoon, a man completed a trip of 375 miles. If the man drove twice as far in the morning as he did in the afternoon, how far did he travel in the morning?

7. A desk and chair together cost $148, and the desk cost three times as much as the chair. Find the cost of the chair; of the desk.

8. The length of a rectangle is 4 in. more than three times the width of the rectangle. If the length of the rectangle is 40 in., find the width of the rectangle.

9. In traveling a distance of 28 mi., a man walked part of the way and rode the rest. If he rode six times as far as he walked, find the distance that he walked.

10. A man has some coins consisting of nickels, dimes, and quarters. He has three times as many dimes as quarters, and twice as many nickels as quarters. How many of each kind of coin has he, if he has 54 coins in all?

11. In his stamp collection a boy had a certain number of United States stamps and four times as many foreign stamp. If there were 12,260 stamps in his collection, how many of these were United States stamps?

12. In a certain algebra class there were three times as many boys as girls. How many boys were there if the class contained 36 students?

Answers : 1. 24 nickels, 12 dimes. 2. daughter = 13 yr., father = 39 yr. 3. 137. 4. Bill is 14 yr. Betty is 7 yr. 5. 18. 6. 250 mi. 7. chair cost $37, desk cost $111. 8. 12 in. 9. 4 mi. 10. 9 quarters, 18 nickels, 27 dimes. 11. 2452 U.S. stamps. 12. 27 boys.

 

CHAPTER REVIEW EXERCISES

Solve each of the equations in ex. 1-10 and check each answer.

11. Twice a certain number diminished by 6 is equal to 16. Find the number.

12. Robert is twice as old as Sue. If the sum of their ages is 39 years, how old is each?

13. One number is 6 more than three times another. If the larger number is 42, what is the smaller number?

14. A man invested some money in bonds, and twice as much money in stocks. If he invested $825 in all, how much was invested in bonds?

15. The length of a rectangle is 4 ft. less than four times the width. If the length of the rectangle is 64 ft., find its width.

Answers : 1. x = 7. 2. x = 13. 3. x = 8. 4. x = 8. 5. x = 2. 6. x = 12. 7. x = 3. 8. x = 9. 9. x = 4. 10. x = 5. 11. 11. 12. Robert is 26 yr., Sue is 13 yr. 13. 12. 14. $275. 15. 17 ft.

CUMULATIVE REVIEW EXERCISES

Answers: 1. 5a. 2. 3b2 3. 2. 4. 0. 5. 1. 6. 6ab. 7. 5x + 5y. 8. 9. 9. 2. 10. 7 + 2y. 11. 3. 12. 42 

13. Perform the indicated multiplication in each of the following by removing parentheses:

a) 5(2x -y)   b) 2x(x2 +x)     c) 4a(2a2 -b2)

14. Express algebraically, "half of the product of x and y."

15. Simplify: 6ab/3a

16. Find the value of the expression 3x2 + 2x -4 when x = 2.

17. How many terms are in the expression 2x -3y + 5?

18. Solve the equation 5x +2x = 18 +17.

19. If 31 is added to three times a number, the result is 85. Find the number.

20. A man has some coins consisting of quarters and dimes. If he has three times as many quarters as he has dimes, and if he has 92 coins in all, how many of each kind of coin has he?

Answers: 13. (a) 10x -5y; (b) 2x3 + 2x2 ; (c) 8a3 -4ab2 . 14. xy/2 . 15. 2b. 16. 12. 17. 3. 18. 5. 19. 18. 20. 23 dimes, 69 quarters.