In the first example we add all the positive quantities, $6+$4 + $8 +$7 = $25. Then we add all the negative quantities, -$5. Adding $25 and $5 the result is $20. - $3 In the second example we add all the positive quantities, and get $6. The sum of the negative quantities is $8. Adding $6 and - $8 the result is $2. Can you give the rule for addition where the quantities have different signs? Which sign does the sum take? Write like quantities in the same column. Find the sum of the positive terms, also the sum of the negative terms; subtract the less from the greater, and prefix the sign of the greater. 3x+14 -7x+9 - 23 4x-5 - 2x 3x+11 7. 4a+3x,-2a, -7x-3a, — 5x, −9a+x. 8.3b+c, 4a+6b, 5b-9c, 3a, -2a-3b+4 c. 9. x8, x+4, -4x-3, 7x+16, -5x-10. 10. 423,8 x + 21, −z x + 11, −x + 5, 9x - 3. SUBTRACTION OF ALGEBRAIC QUANTITIES. 541. Preliminary Exercises. 1. A man sold a horse for $100 at a gain of $25. Find the cost. (Cost = selling price — gain.) subtract remainder $ 75= $100 or add - 25 $ 75 2. A man sold a horse for $100 at a gain of Find the cost. $100 = selling price $25. $100 or add + 25 $125 $125 = cost In the first of the above examples, subtracting + $25 is the same as adding $25. In the second of the above examples, subtracting - $25 is the same as adding + $25. We changed the first example from subtraction to addition by changing the sign of the subtrahend from + to We changed the second example from subtraction to addition by changing the sign of the subtrahend from To subtract in algebra, change the sign of the subtrahend and proceed as in addition. 9. Subtracting — 7 is the same as adding what? - 10. Is a positive quantity increased or decreased by subtracting a negative quantity? When you become familiar with the process of subtraction it will not be necessary to write the subtrahend with a changed sign. You can conceive the sign changed and add. 542. Sight Exercises. 1. What is the difference between + 52° and +33° ? 2. Between + 90° and -10°? Show by a diagram. 3. A has $600, B owes $400. What are they worth together? (+$600)+(− $ 400) = ? 4. How much better off is A than B? (+$600) − (− $ 400) = ? 16. From 7y-2z+b take 8y+66-2%. 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: 6x+15y (4 x + 10 y). 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 4 x? 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? a + (b −c) 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. 2. 92 (63+25) = 4. 4. (178) (1614) = 7. 5. 75+4 x (1510) = 95. 3. (43-10)+(24-5)=52. 6. 75-4 × (15—10) = 55. 7. 4x+5y+ (2 x − 6 y) = 6 x — y. 8. 4x+5y-(2x+6y)=2x-y. 10. 4x-5y-(-x+6 y) = 5x-11 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. 2. 7(x+2)=3x+50. 4. 3(16x)=4(13x). Bringing to the right side of the equation, and - 27 to the left side, we have (+)27 = (+)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 = Clearing of fractions, 15 x 30 – (8 x − 8) = 20. |