REMOVING PARENTHESES. 544. Written Exercises. 1. From 6x + 15 y take 4 x + 10 y. We may write the above in a shorter way, thus: The minus sign before the parenthesis shows that the quantity within the parenthesis is to be subtracted. What sign is before 10 y? What sign is understood within the parenthesis before 4x? In subtraction, what is done with. the signs of the subtrahend? If the whole expression is written without using the parenthesis, what must be done with the signs of the quantities within the parenthesis? a+ (bc) may be written a+b-c. Why? When removing a parenthesis preceded by a minus sign, change the signs of all quantities within the parenthesis. 545. Written Exercises. Write the following without parentheses: 1. 57 +(3316) = 74. 4. (178) (16 — 14) = 7. 5. 75+4 × (15 — 10) = 95. × 3. (43-10)+(245) = 52. 6. 75-4 x (15-10)= 55. 7. 4x+5y+ (2 x − 6 y) = 6 x − y. 12. -4x-5y + (2 x − 6 y) = ?. 546. Solve the following equations. Prove the correctness of your answers. 1. 6(2x-5)=5x+12. NOTE. 6(2x-5) means 6 times (2 x 5), or 12 x — 30. -- 38 11+9x= = 10x. 9x10x — 38 + 11, == -x=-27. x to the right side of the equation, and 27 to the left (+)27 = (+)x. x= In practice, however, when the result is such as the above, - 27, the signs of both members are changed, and the result becomes Clear of fractions by multiplying both members of the equation by 10, and observe which sign must be changed to preserve the equality. When x = 6, the above may be written Removing the parenthesis, Transposing, or, NOTE. The horizontal line between the numerator and the denominator of the foregoing fractions has the effect of a parenthesis, the entire quantity above the line being divided by the number below. Hence when an equation is cleared of fractions, what must be done with the signs of the terms obtained from a fraction with a minus sign ? 24 - 4 = 3 of (24 — 4). 6 18 = (186) ÷ 2, 5 4x 4 5 1. A certain number is multiplied by 3; 7 is subtracted from the product; the remainder is divided by 16, giving a quotient of 3. What is the number? 2. Three-eighths of what number is 60 less than the number itself? 3. Four persons are of the of his age older, the second of his age older, and the fourth of their ages would be 99 years. same age. 4. A man spends of his earnings on board and lodging, on clothing and repairs, and on sundries. At the end of the year he has $280 left. What are his yearly earnings? 5. A boy gave of his marbles to one companion, and of them to another. He then bought as many as he originally had, and had 4 marbles more than he had at first. How many did he have at first? 6. A father's age and a son's age added together amount to 138 years. Twelve years ago the father was twice as old as the son. How old is each now? Let x = son's age 12 years ago. 2x father's age then. How 7. John has 80 cents, and William has 60 cents. many cents should William give John so that the latter might have 21⁄2 times as much money as the former? After William gives John x cents, the former has (60 - x) cents, and the latter has (80+x) cents. 8. In how many years will a man, now 25, be double the age of his 11-year-old brother? Let x number of years. 25 +x and 11+x=ages after x years. 9. A man has a cask of 60 gallons' capacity. He draws off one-fourth of its contents, and then fills it. If it takes 24 gallons to fill it, how many gallons did the cask originally contain? 10. A number is divided by 3, and 40 is subtracted from the quotient, leaving a remainder of 104. What is the number? 11. The difference between two numbers is 430. When the greater is divided by the less, the quotient is 4, and the remainder is 76. What are the numbers? 12. A person pays $103 with 29 $2 and $5 bills. How many are there of each denomination ? 13. A father is 30 years older than his daughter. 4 years, his age will be four times her age. present ages? In What are their ages 4 years later. x and x+30= present ages. x+4 and x + 34. 14. The product of two numbers is 180. If the smaller number be increased by 3, the product of the two numbers will be 225. What are the numbers? 15. A man's wages are $1 per day more than his son's. For 33 days' work, the father receives $12 more than the son earns in 40 days. Find the wages of each. |