2 Making sense of algebra
Note that a ‘+’ or a ‘−’ that appears within an algebraic expression, is attached to the term that sits to its right. For example: 3x − 4y contains two terms, 3x and −4y. If a term has no symbol written before it then it is taken to mean that it is ‘+’.
Worked example 6 Simplify: (a) 4a + 2a + 3a (d) 2p + 5q + 3q − 7p
3 x 2y 2y + 5 z = 3 5 z − 2y =5 3 x − 2y = −2 y 3x 3x + 5z
4a 2a 2a + 3a = 9a
Terms are all like. Add the coefficients, write the term.
(b)
4a
6 b + 3a
Identify the like terms (4a and 3a). Add the coefficients of like terms. Write terms in alphabetical order.
6b
(c)
5 x 2y 2y − 7 x = −2 x 2 y
(d)
2p 5 5q q + 3q = 5 p 8q
(e)
2 b
Identify the like terms (5x and −7x). Subtract the coefficients, remember the rules. Write the terms. (This could also be written as 2y − 2x.) Identify the like terms (2p and −7p; 5q and 3q). Add and subtract the coefficients. Write the terms.
7p
3a 3a2 b − ab + 3ab2
= ab + 3
Exercise 2.3
(c) 5x + 2y − 7x
(a)
= 7a
Notice that you can rearrange the terms provided that you remember to take the ‘−’ and ‘+’ signs with the terms to their right. For example:
(b) 4a + 6b + 3a (e) 2ab + 3a2b − ab + 3ab2
2
3ab
Identify like terms; pay attention to terms that are squared because a and a2 are not like terms. Remember that ab means 1ab.
2
1 Identify the like terms in each set. (a) 6x, −2y, 4x, x (d) 2, −2x, 3xy, 3x, −2y
(b) x, −3y, 34 y, −5y (e) 5a, 5ab, ab, 6a, 5
(c) ab, 4b, −4ba, 6a (f) −1xy, −yx, −2y, 3, 3x
2 Simplify by adding or subtracting like terms. (a) 2y + 6y (d) 21x + x (g) 9x − 10x (j) 9xy − 2xy (m) 4x2 − 2x2 (p) 14ab2 − 2ab2 FAST FORWARD
You will need to be very comfortable with the simplification of algebraic expressions when solving equations, inequalities and simplifying expansions throughout the course. X
(b) (e) (h) (k) (n) (q)
9x − 2x 7x − 2x y − 4y 6pq − 2qp 9y2 − 4y2 9x2y − 4x2y
(c) (f) (i) (l) (o) (r)
10x + 3x 4y − 4y 5x − x 14xyz − xyz y2 − 2y2 10xy2 − 8xy2
(b) (e) (h) (k) (n) (q)
4y − 2y + 4x 4xy − 2y + 2xy 3y + 4x − x 12x2 − 4x + 2x2 xy − 2xz + 7xy 4xy − x + 2yx
(c) (f) (i) (l) (o) (r)
6x − 4x + 5x 5x2 − 6x2 + 2x 4x + 6y + 4x 12x2 − 4x2 + 2x2 3x2 − 2y2 − 4x2 5xy − 2 + xy
3 Simplify: (a) 2x + y + 3x (d) 10 + 4x − 6 (g) 5x + 4y − 6x (j) 9x − 2y − x (m) 5xy − 2x + 7xy (p) 5x2y + 3x2y − 2xy 4 Simplify as far as possible: (a) 8y − 4 − 6y − 4 (d) y2 + 2y + 3y − 7 (g) 4xyz − 3xy + 2xz − xyz
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Unit 1: Algebra
(b) x2 − 4x + 3x2 − x (e) x2 − 4x − x + 3 (h) 5xy − 4 + 3yx − 6
(c) 5x + y + 2x + 3y (f) x2 + 3x − 7 + 2x (i) 8x − 4 − 2x − 3x2