Algebra

1. Algebra

2. Collecting Terms • This is a way of simplifying algebra • If you have b + b + b + b • This is the same as 4 b • The b could stand for boots • If you have p x p x p • This is p3 • This happens when multiplying

3. Try these: • M x m • T + t + t • R x r x r x r • G + g + g + g • H x h x h • K + k • J x j x j x j x j x j x j

4. Answers • M2 • 3t • R4 • 4g • H3 • 2k • j7

5. Collecting terms • Usually you have to collect terms from a mix • You cannot add 2t and t2, you can only add t2 and t2 together • E.g. • A2 + 3a + 3a + 2a2 • 3a + 3a = 6a • A2 + 2a2 = 3 a2 • So the simplified equation is 6a + 3 a2

6. Collecting terms • You may have to deal with minus numbers or terms • Always look at the sign before the term to see if it is positive (+) or negative (-) • E.g. 3 -1 -4 • This is +3 and -1 and -4 • 3 -1 = 2 • 2 - 4 = -2

7. Now try these; • -5 +4 -3 • 3h + 2h – 6h • 3s + 3s – 8s • 7y – 5y – 2y • 3d – d + 2d – 5d • 7r – 4r – 5r -2r • 5p – p + 3p – 2p

8. Answers • -4 • -1h or -h • -2s • 0 • -1d or –d • -4r • 5p

9. Mixed terms • J + j + k + k + k • R – 2r +3s +2s • 7y – 5y – 3y +4 • 3t + 6s – 8s + t • 7r + 2s – r – s • 5p + 6 – 7p – 9 • 3a + b + a - 2b

10. Answers • 2j + 3k • -r + 5s • -y + 4 • 4t – 2s • 6r + s • -2p – 3 • 4a - b

11. Multiplying Terms • When you multiply terms you multiply the numbers at the start of the term and then add together the number of letters you have • E.g. • 2a x 3a • This is 2 x 3 = 6 • And a x a = a2 • This is 6a2

12. Multiplying • 2 x 4c • 2 x r x s • 5 x 3c • 2e x 6e • 3a x 2a • (4k)2 • 4r x 2rs

13. Answers • 8c • 2rs • 15c • 12e2 • 6a2 • 16k2 • 8r2s

14. Try these • 5ab – 3ab • 6vw – 4w + 5wv • X + 2x – 3x2 + 5x2 • 8r + 6rs – 2sr – 3r • Xy + x2 – 3xy + 3x2 • 4y + 3y2 – 7y2 – 2y

15. Answers • 2ab • 11 vw + 4w • 3x + 2x2 • 5r – 4rs • -2xy + 4x2 • 2y – 4y2

16. Multiplying Out Brackets • Everything inside the brackets is multiplied by the term just outside it to the left • E.g. • 4 (3 + t) • This is • 4 x 3 = 12 • And 4 x t= 4t • So it becomes 12 + 4t

17. Try these • 6(1-s) • 4(p + q) • 3(10j-4k) • R2(3-2s) • 2x3 (x-y) • 5t2 (s + t) • 3r (2r – 3s – t)

18. Answers • 6 -6s • 4p + 4q • 30j + 12k • 3r2 + -2r2s • 2x4 – 2x3y • 5st2 + 5t3 • 6r2 – 9rs – 3rt

19. Factorising • This is the opposite of multiplying out brackets • When you simplify you can place terms inside brackets • E.g. 4k + 2 • Both can be divided by 2 • So it becomes 2(2k + 1) • Everything inside the bracket is divided by 2

20. Try these • 3f + 3 • 15 + 20 t • 18 + 6a • 10j + 25 • 3r + 3s + 3t • Pq – q2 • 24p2 + 30pq • 20ab2 + 36a2b2

21. Answers • 3 (f + 1) • 5 (3 + 4t) • 6 (3 + a) • 5 (2j + 5) • 3 (r + s + t) • Q (p – q) • 6p (4p + 5q) • 4ab (5b + 9ab)

22. Multiplying Brackets with - • If there is a minus before the brackets • A minus x a plus = a minus • A minus x a minus = a plus • E.g. • -3(2r – r) • -3 x 2r= -6r • -3 x –r = +3r • So it becomes -6r + 3r

23. Try these • -8(s-t) • -(r-5) • -(4r-3) • -9y(y-1) • -5s(s+4) • -3h(5-h) • - (x+y)

24. Answers • -8s + 8t • -r + 5 • -4r + 3 • -9y2 + 9y • -5s2 + 20s • -15h + 3h2 • -x - y

25. Multiplying sets of brackets • If you are given 2(3+y) + 5(4+y) • First you must multiply out the brackets • 6 + 2y + 20 + 5y • Secondly you collect terms • 26 + 7y

26. Try these • 3(4+d)+4(2+d) • 6(3+x)+5(2-x) • 2(10+5e)-3(6+e) • 3(4r+1)-(7r-2) • 4(w+1)-(w-1) • X(2x+1)+2(3x+4) • 4(5+2f)+f(3+f)

27. Answers • 20 + 7d • 28 + x • 2 + 7e • 5r + 5 • 3w + 5 • 2x2 + 7x + 8 • 20 + 11f + f2

28. Multiplying out brackets • If you are given (y+2)(y-4) • Then you really have two sums • Y(y-4) • +2 (y-4) • We have split the first bracket up to make sure we multiply everything together • So what do they workout as; • Y2 – 4y • +2y -8 • We need to collect the terms • -4y + 2y = -2y • So the equation is y2 -2y -8

29. Try these • (s+1)(3s+2) • (2+f)(1+4f) • (d-2)(3d+5) • (7+k)(1+k) • (a+3)(4a-1) • (y-2)(y+2) • (a+b)(a-b)

30. Answers • 3s2+5s+2 • 2+9f+4f2 • 3d2-d-10 • 7+8k+k2 • 4a2+11a-3 • Y2-4 • A2-b2

31. (Brackets)2 • If you have (4g+h)2 • This means (4g+h) )(4g+h) • First we … • Split up the first bracket • 4g(4g+h) • +h(4g+h) 2. Then we multiply this out • 16g2 + 4gh • +4gh + h2 3. Then we collect terms • +4gh + 4gh = +8gh • So our equation is • 16g2 + 8gh + h2

32. Solving Equations • If you have a number and no x2 or x3 you can solve a linear equation • E.g. • 4x=16 • X = 16/4 • X= 4 • x/7=-2 • Multiply both sides by 7 • X = -14 • What you do to one side of the equals sign you must do to the other to keep everything balanced

33. Try these • x/5= 12 • X-7=23 • X + 12 = 45 • 0.5x=3 • 2x/3 = -6 • x/3 + 2 =10 • 7x= x + 42 • 2.5x = 1.5x + 6

34. Answers • X= 60 • X= 30 • X= 33 • X= 6 • X= -9 • X= 24 • X= 7 • X= 6

35. Solving Equations • You should always try to keep x positive, so work from the side with the most x’s • E.g. • 2x + 17= 4x • We need to take 2x away from both sides • 17 = 2x • 8.5 = x

36. Harder questions • 13x-15 = 12x+19 • 2.5x + 10 = 1.5x + 17 • 12x-1 = 7x+19 • 4x-3 = 12-x • 2x-7 = 3 – 8x • 3(2x+2) = 24 • 6(3x+1) = 3(4x+6) • 10(3x-4) = 5(6-x)

37. Answers • X= 34 • X= 7 • X= 4 • X= 3 • X= 1 • X= 3 • X= 1 • X= 2

38. Changing the Subject • Sometimes you have to rearrange equations • E.g. • C= a – b • I want to find out what a equals • If I add b to both sides I have a on it’s own • C + b = a

39. Try these • T= sk - h • V=u + at • M=k + nk • T2 = 7r + x/4 • H= y/4 • T= s2 + 5 • A=πr2 • S= ½gt2

40. Answers • T/s + h/s = k • v/a – u/a = t • m/k– k/k = n • T2/7- x/4/7 = r • 4h = y • √t-5 = s • √a/ π = r • √2s/g= t