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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Brackets

Quantities are enclosed within brackets to indicate that they are to be treated as a single entity. If we wish to subtract, say, 3a - 2b from 4a - 5b then we do this as follows.

Example 1

(a)

 (4a - 5b ) - (3a - 2b ) = 4a - 5b - 3a - ( -2b ) = 4a - 5b - 3a + 2b = 4a - 3a - 5b + 2b = a - 3b

and similarly

(b)

 (7 x + 5 y ) - (2 x - 3 y ) = 7 x + 5 y - 2 x - ( - 3 y ) = 7 x + 5 y - 2 x + 3 y = 7 x - 2 x + 5 y + 3 y = 5 x + 8 y .

When there is more than one bracket it is usually necessary to begin with the inside bracket and work outwards.

Example 2

Simplify the following expressions by removing the brackets.

(a) 3 a - c + (5 a - 2 b - [3 a - c + 2 b]),

(b) -{3 y - (2 x - 3 y) + (3 x - 2 y)} + 2 x.

Solution

(a) We have

 3 a - c + (5 a - 2 b - [3 a - c + 2 b]) = 3 a - c + (5 a - 2 b - 3 a + c - 2 b) = 3 a - c + (2 a - 4 b + c) = 3 a - c + 2 a - 4 b + c = 3 a + 2 a - 4 b - c + c = 5 a - 4 b .

(b) Similarly we have

 -3 y - (2 x - 3 y) + (3 x - 2 y) + 2 x = -3 y - 2 x + 3 y + 3 x - 2 y + 2 x = -3 y + 3 y - 2 y + 3 x - 2 x + 2 x = -4 y + x + 2 x = -4 y - x + 2 x = x - 4 y .

Exercise

Remove the brackets from each of the following expressions and simplify as far as possible.

(a) x - ( y - z ) + x + ( y - z ) - ( z + x ) ,

(b) 2 x - (5 y + [3 z - x] ) - (5 x - [ y + z ] ),

(c) (3 /a) + b + (7 /a) - 2 b,

(d) a - ( b + [ c - {a - b}] ) .

Solution

(a)

x - ( y - z ) + x + ( y - z ) - ( z + x )

= x - y + z + x + y - z - z - x

= x + x - x - y + y + z - z - z

= x - z .

(b)

2 x - (5 y + [3 z - x ]) - (5 x - [ y + z ])

= 2 x - (5 y + 3 z - x ) - (5 x - y - z )

= 2 x - 5 y - 3 z + x - 5 x + y + z

= 2 x + x - 5 x - 5 y + y - 3 z + z

= - 2 x - 4 y - 2 z .

(c)

(d)

a - ( b + [ c - { a - b } ])

= a - ( b + [ c - a + b ])

= a - ( b + c - a + b )

= a - (2 b + c - a )

= a - 2 b - c + a

= 2 a - 2 b - c .