Algebra

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Algebra - PowerPoint PPT Presentation

Algebra. 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 p 3 This happens when multiplying. Try these:. M x m T + t + t R x r x r x r G + g + g + g H x h x h

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Algebra

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
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
• M2
• 3t
• R4
• 4g
• H3
• 2k
• j7
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
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
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
• -4
• -1h or -h
• -2s
• 0
• -1d or –d
• -4r
• 5p
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
• 2j + 3k
• -r + 5s
• -y + 4
• 4t – 2s
• 6r + s
• -2p – 3
• 4a - b
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
Multiplying
• 2 x 4c
• 2 x r x s
• 5 x 3c
• 2e x 6e
• 3a x 2a
• (4k)2
• 4r x 2rs
• 8c
• 2rs
• 15c
• 12e2
• 6a2
• 16k2
• 8r2s
Try these
• 5ab – 3ab
• 6vw – 4w + 5wv
• X + 2x – 3x2 + 5x2
• 8r + 6rs – 2sr – 3r
• Xy + x2 – 3xy + 3x2
• 4y + 3y2 – 7y2 – 2y
• 2ab
• 11 vw + 4w
• 3x + 2x2
• 5r – 4rs
• -2xy + 4x2
• 2y – 4y2
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
Try these
• 6(1-s)
• 4(p + q)
• 3(10j-4k)
• R2(3-2s)
• 2x3 (x-y)
• 5t2 (s + t)
• 3r (2r – 3s – t)
• 6 -6s
• 4p + 4q
• 30j + 12k
• 3r2 + -2r2s
• 2x4 – 2x3y
• 5st2 + 5t3
• 6r2 – 9rs – 3rt
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
Try these
• 3f + 3
• 15 + 20 t
• 18 + 6a
• 10j + 25
• 3r + 3s + 3t
• Pq – q2
• 24p2 + 30pq
• 20ab2 + 36a2b2
• 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)
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
Try these
• -8(s-t)
• -(r-5)
• -(4r-3)
• -9y(y-1)
• -5s(s+4)
• -3h(5-h)
• - (x+y)
• -8s + 8t
• -r + 5
• -4r + 3
• -9y2 + 9y
• -5s2 + 20s
• -15h + 3h2
• -x - y
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
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)
• 20 + 7d
• 28 + x
• 2 + 7e
• 5r + 5
• 3w + 5
• 2x2 + 7x + 8
• 20 + 11f + f2
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
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)
• 3s2+5s+2
• 2+9f+4f2
• 3d2-d-10
• 7+8k+k2
• 4a2+11a-3
• Y2-4
• A2-b2
(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
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
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
• X= 60
• X= 30
• X= 33
• X= 6
• X= -9
• X= 24
• X= 7
• X= 6
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
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)
• X= 34
• X= 7
• X= 4
• X= 3
• X= 1
• X= 3
• X= 1
• X= 2
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
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