We need not, however, adhere strictly to this rule, for the terms may be added or subtracted in the order we find most convenient.
This process is called collecting terms.
Ex. 3. Find the sum of 2 3 a , 3 a , − 1 6 a , − 2 a {\displaystyle {\tfrac {2}{3}}a,3a,-{\tfrac {1}{6}}a,-2a} .
= 3 2 3 a − 2 1 6 a = 1 1 2 = 3 2 a {\displaystyle =3{\tfrac {2}{3}}a-2{\tfrac {1}{6}}a=1{\tfrac {1}{2}}={\tfrac {3}{2}}a}
Note. The sum of two quantities numerically equal but with opposite signs is zero. The sum of 5 a {\displaystyle 5a} and − 5 a {\displaystyle -5a} is 0.
EXAMPLES II.
Find the sum of
1. 5 a , 7 a , 11 a , a , 23 a {\displaystyle 5a,7a,11a,a,23a} . 2. 4 x , x , 3 x , 7 x , 9 x {\displaystyle 4x,x,3x,7x,9x} . 3. 7 b , 10 b , 11 b , 9 b , 2 b {\displaystyle 7b,10b,11b,9b,2b} . 4. 6 c , 8 c , 2 c , 15 c , 19 c , 100 c , c {\displaystyle 6c,8c,2c,15c,19c,100c,c} . 5. − 3 x , − 5 x , − 11 x , − 7 x {\displaystyle -3x,-5x,-11x,-7x} . 6. − 5 b , − 6 b , − 11 b , − 18 b {\displaystyle -5b,-6b,-11b,-18b} . 7. − 3 y , − 7 y , − y , − 2 y , − 4 y {\displaystyle -3y,-7y,-y,-2y,-4y} . 8. − c , − 2 c , − 50 c , − 13 c {\displaystyle -c,-2c,-50c,-13c} .
9. − 11 b , − 5 b , − 3 b , − b {\displaystyle -11b,-5b,-3b,-b} . 10. 5 x , − x , − 3 x , 2 x , − x {\displaystyle 5x,-x,-3x,2x,-x} . 11. 26 y , − 11 y , − 15 y , y , − 3 y x , 2 y {\displaystyle 26y,-11y,-15y,y,-3yx,2y} . 12. 5 f , − 9 f , − 3 f , 21 f , − 30 f {\displaystyle 5f,-9f,-3f,21f,-30f} . 13. 2 s , − 3 s , s , − s , − 5 s , 5 s {\displaystyle 2s,-3s,s,-s,-5s,5s} . 14. 7 y , − 11 y , 16 y , − 3 y , − 2 y {\displaystyle 7y,-11y,16y,-3y,-2y} . 15. 5 x , − 7 x , − 2 x , 7 x , 2 x , − 5 x {\displaystyle 5x,-7x,-2x,7x,2x,-5x} . 16. 7 a b , − 3 a b , − 5 a b , 2 a b , a b {\displaystyle 7ab,-3ab,-5ab,2ab,ab} .
Find the value of
17. − 9 x 2 + 11 x 2 + 3 x 2 − 4 x 2 {\displaystyle -9x^{2}+11x^{2}+3x^{2}-4x^{2}} . 18. 3 a 2 x − 18 a 2 x + a 2 x {\displaystyle 3a^{2}x-18a^{2}x+a^{2}x} .
19. 3 a 3 − 7 a 3 − 8 a 3 + 2 a 3 − 11 a 3 {\displaystyle 3a^{3}-7a^{3}-8a^{3}+2a^{3}-11a^{3}} . 20. 4 x 3 − 5 x 3 − 8 x 3 − 7 x 3 {\displaystyle 4x^{3}-5x^{3}-8x^{3}-7x^{3}} .
21. 4 a 2 b 2 − a 2 b 2 − 7 a 2 b 2 + 5 a 2 b 2 0 a 2 b 2 {\displaystyle 4a^{2}b^{2}-a^{2}b^{2}-7a^{2}b^{2}+5a^{2}b^{2}0a^{2}b^{2}} . 22. − 9 x 4 − 4 x 4 − 12 x 4 + 13 x 4 − 7 x 4 {\displaystyle -9x^{4}-4x^{4}-12x^{4}+13x^{4}-7x^{4}} . 23. 7 a b c d − 11 a b c d − 41 a b c d + 2 a b c d {\displaystyle 7abcd-11abcd-41abcd+2abcd} . 24. 1 2 x − 1 3 x + x 2 3 x {\displaystyle {\tfrac {1}{2}}x-{\tfrac {1}{3}}x+x{\tfrac {2}{3}}x} .25. 3 2 a + 3 5 a − 1 2 a {\displaystyle {\tfrac {3}{2}}a+{\tfrac {3}{5}}a-{\tfrac {1}{2}}a} . 26. − 5 b + 1 4 b − 3 2 + 2 b − 1 2 b + 7 4 b {\displaystyle -5b+{\tfrac {1}{4}}b-{\tfrac {3}{2}}+2b-{\tfrac {1}{2}}b+{\tfrac {7}{4}}b} . 27. − 5 3 x 2 − 2 x 2 − 2 3 x 2 + x 2 + 1 2 x 2 + 11 6 x 2 {\displaystyle -{\tfrac {5}{3}}x^{2}-2x^{2}-{\tfrac {2}{3}}x^{2}+x^{2}+{\tfrac {1}{2}}x^{2}+{\tfrac {11}{6}}x^{2}} . 28. − a b − 1 2 a b − 1 3 a b − 1 4 a b − 1 6 a b + 5 12 a b {\displaystyle -ab-{\tfrac {1}{2}}ab-{\tfrac {1}{3}}ab-{\tfrac {1}{4}}ab-{\tfrac {1}{6}}ab+{\tfrac {5}{12}}ab} . 29. 2 3 x − 3 4 x 2 − 5 6 x − 2 x + 11 6 x − 1 3 x + x {\displaystyle {\tfrac {2}{3}}x-{\tfrac {3}{4}}x^{2}-{\tfrac {5}{6}}x-2x+{\tfrac {11}{6}}x-{\tfrac {1}{3}}x+x} . 30. − 5 3 x 2 − 3 4 − 4 3 x 2 − 1 4 x 2 − x 2 {\displaystyle -{\tfrac {5}{3}}x^{2}-{\tfrac {3}{4}}-{\tfrac {4}{3}}x^{2}-{\tfrac {1}{4}}x^{2}-x^{2}} .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
