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The A to Z of mathematics a basic guide - Sidebotham T.H.

Sidebotham T.H. The A to Z of mathematics a basic guide - Wiley publishing , 2002. - 489 p.
ISBN 0-471-15045-2
Download (direct link): theatozofmath2002.pdf
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= —xy When the coefficient of a term is 1 or — 1, it is not written with the term
12 ALGEBRA
For part (ii)
a + 2b — 2a + ab = a — 2a + 2b + ab Grouping the like terms together = — a + 2b + ab There are no more like terms
Example 3. Now find the area of the house.
Solution. To find the area of the ground floor of the house, we need to divide the shape up into two rectangles, find the area of each, and add them together (see figure b):
x
3x
3y 8

18
(b)
Area of larger rectangle = length x width
= 3x x x
= 3x2 square meters x2 is shorthand for x x x Area of the smaller rectangle = length x width
= 8 x y
= 8y square meters 8y is shorthand for 8 x y Total area of the house = sum of areas of the two rectangles = 3x2 + 8y
This expression cannot be simplified, because the two terms are not like terms.
In the example of finding the area of the house we multiplied terms together. We now study some more examples that explain how to simplify expressions when terms are multiplied.
If the length and width of a square are x, then
Area of a square = length x width
A = x • x The dot may be used for “times”
A = x2 In words we say “x squared”
ALGEBRAIC FRACTIONS 13
If the length, width, and height of a cube are all x, then
Volume of a cube = length x width x height
V = x • x • x
V = x3 In words we say x cubed”
Example 4. Simplify these expressions: (i) 2 x 3x, (ii) 2ab x 4b
Solution. For part (i)
2x3.x = 2x3xi Inserting the x sign between 3 and x
= 6 x x Multiplying the numbers 2 and 3 first
= 6x Writing without the x sign
For part (ii)
2 ab x4b = 2 ■ a ■ b ■ 4 ■ b Inserting dots for the x signs
= 2 • 4 • a • b • b Grouping terms in alphabetical order with numbers
first.
= 8 • a -b2 b-b = b2
= 8 ab2
The process of dividing algebraic terms and simplifying the answer, is explained under the entry Canceling.
References: Abstract, Algebraic Equations, Coefficient, Equations, Factor, Indices, Substitution, Variable.
ALGEBRAIC EQUATIONS
References: Balancing an Equation, Equations, Linear Equation, Quadratic Equations, Simultaneous Equations, Solving an Equation.
ALGEBRAIC FRACTIONS
Algebraic fractions, like arithmetic fractions, can be canceled, added, subtracted, multiplied, and divided. Canceling fractions is explained under the entry Canceling. It may be helpful to realize that if two fractions are equivalent, they can be written differently, as demonstrated here.
When the variable y is multiplied by the fraction | the result is |y. Also, when 5 divides y, the result is written as I.
14 ALGEBRAIC FRACTIONS
Since multiplying by | achieves the same result as dividing by 5, it follows that |y and | are equivalent. Similarly, \x and are equivalent.
In order to add and subtract fractions they must have the same denominator. For example,
X y X + y
— + — = —-— The two fractions have the same denominator, 7
111
If the denominators are not the same, each fraction has to be converted to an equivalent fraction so that the denominators are the same size.
Example. Simplify
x 2x 5 + T
Solution. The two denominators are 5 and 3. The lowest common multiple of 5 and 3 is the lowest positive number they both divide into exactly, which is 15. When working with fractions the lowest common multiple is called the lowest common denominator. Each fraction is then written with a denominator of 15, using equivalent fractions, as set out here:
x 3 3.x 2x 5 IOjc
— x - = — and — x - = ——
5 3 15 3 5 15
Now,
x 2x 3x lOx
— H = —- H — Replacing each fraction in the sum by its equivalent
5 3 15 15 r j •
fraction
3.x + lOx
15
The denominators are the same and numerators are added
13x
u
Which is a single fraction Example. Simplify this expression
x2 xy
by subtracting the two fractions.
ALGEBRAIC FRACTIONS 15
Solution. The denominators are x2 and xy. The lowest common multiple of x2 and xy isa;2y. Thus
2 3 2 y 3 x
x2 xy x2 y xy x
2 y 3x
x2y x2y
2 y — 3x
x2y
Replacing each fraction by its equivalent fraction
Simplifying each fraction
The denominators are the same and fractions are subtracted
Two fractions can be multiplied to obtain a single fraction. The numerators are multiplied together, and the denominators are multiplied together.
Example. Simplify
2x 5x Y X ~4
Solution. Write
2x 5x 2x x 5x
3x4
10a:2
12
5a;2 2
~~6~ X 2
5a:2
2 is a common factor of numerator and denominator
Canceling the 2’s
Two fractions can be divided to obtain a single fraction. The method is explained in the following example.
Example. Simplify
3a; 1
T “ 2y
by dividing the fractions.
16 ALGEBRAIC FRACTIONS
Solution. Write
— ^ — x — Instead of dividing, multiply by the reciprocal of the
^ second fraction
Multiplying the numerators and denominators
= x - 2 is a common factor
1 2
= 3a:y Canceling the 2’s
Example. On his way to work, in the car, David reckons his speed through the traffic is 30 kilometers per hour, abbreviated km h-1, whereas for the return journey it is 40 km h-1. Calculate his average speed.
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