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DCUSchoolof Mathematical Sciences BASIC SKILLS by mbw16088

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									                            DCU School of Mathematical Sciences
                               BASIC SKILLS WORKSHEET 3
                            Algebra I - Variables and Expressions
    The aim of this worksheet is to revise some maths that deals with manipulating formulas and
expressions. This is the basic ‘grammar’ of maths that allows us to use mathematical language in
science, business and engineering.

What is algebra?
This is the name given to the area of maths that deals with fiddling about with symbols and letters.

What is it for?
It’s for making our lives easier. Up until the 14th Century or so, people used to write things like this:

  “It is desired that quantity for which thrice the quantity added to twice its own product has the
                                          magnitude twelve.”

It’s easier to say “Find x where 2x2 + 3x = 12.”

What is a variable?
A variable is a symbol (usually a letter) that we use to represent a quantity whose value can change
or whose value is not known. For example,

                                   P = price of a gallon of petrol
                                    x = length of a piece of string
                                y = amount of salt in a saline solution

It is also common to use letters to represent quantities whose values are fixed and are known. This
is very common in science, where examples include

                                   c = speed of light in a vacuum
                                 G = Newton’s gravitational constant

Letters or symbols used in this way are called constants. We also use the word constant to describe
                      √
fixed numbers, e.g. 3, 2, 1 , etc.
                          2


What is an expression?
This means a collection of variables and numbers combined using addition, subtraction, multiplication
etc. For example, if x and y represent unknown numbers, then the following are expressions involving
x and y:
                                                               2x − y
                              2x + y,      xy,    3x2 + 8y,           .
                                                               x + 4y


                                                   1
Reminder
In order to save time and ink, we use these shorthand ways as substitutes for × (multiplication) and
÷ (division):


                 two quantities written together means multiplication: xy = x × y;
        two quantities written above and below a horizontal or diagonal bar means division:
                                         x
                                           = x/y = x ÷ y.
                                         y
Here are two more definitions that will be useful to have.

Terms
A term is an individual part of an expression. For example the expression

                                        2x2 − 4xy + 18y 2 + 21

contains four terms; 2x2 , −4xy, 18y 2 and 21. These are called respectively “the term in x2 ”, “the
term in xy” and “the term in y 2 ”. The term which is just a number is called the “constant term”.

Coefficients
A coefficient is a quantity (usually a number or a constant) that multiplies another quantity. For
example, in the expression
                                       −2ab + 4c2 + 11bc
the coefficient of ab is -2; the coefficient of c2 is 4 and the coefficient of bc is 11.

Equations
An equation is simply a statement that two expressions are equal to one another. For example,


                                                x=4
                                           x2 − 3x + 2 = 0
                                      −2ab + 4c2 + 11bc = 15b
                                              E = mc2

The difference between expressions and equations is that expressions never contain an equals sign.




                                                  2
Brackets
Suppose that we needed to multiply the expression x − 2 by 3. The result could be written as
                                             3 × x − 2.
However we don’t usually use the × sign. Also, we need a way of indicating that all of the expression
x − 2 is being multiplied by 3. We use brackets to do this:
                                       3 × x − 2 = 3(x − 2).
This means that every term in the brackets gets multiplied by 3, and so the result is
                                         3(x − 2) = 3x − 6.

Exercise 1
  1. In the following expressions, write down the required quantity.
      (a) 2x2 − 4x + 3         write down the coefficient of x.
                   2
      (b) 3bc + 5b − abc         write down the term in bc.
      (c) −P 2 + 4P − 32          write down the constant term.
  2. A major use of algebra is for writing down rules about e.g. science, economics, engineering in
     a concise way. For example, consider the total revenue generated by selling a certain number
     of cars at a certain price. Would could write

                           total revenue = number of cars × price of each car.

     However it makes life a bit easier if we just abbreviate this by using some definitions. Let



                                            T = total revenue,
                                            P = price of car,
                                    Q = quantity (or number) of cars.

     Then we get the simple equation
                                                T = P Q.
     An equation like this which must always be true for the variables T, P, Q is called a formula.
     In the following examples, invent names (single letters) for the relevant quantities, and write
     down the formula which is described in words.
      (a) The average speed of a car is the distance travelled divided by the time taken.
      (b) The area of a rectangle is the length multiplied by the breadth.
      (c) Temperature in Farenheit is found by multiplying temperature in Celsius by 1.8 and then
          adding 32.

                                                 3
Substituting in expressions and formulae
  • This means replacing variables with actual numbers. For example consider the following ex-
    pressions in u and v:



                                                   u+v

                                                   u−v

                                                  2u2 + v 2
                                                  2u − v
                                                         .
                                                  v + 3u

    If u = 2, v = 3 then these expressions take the values



                                                 2+3=5

                                                 2 − 3 = −1

                                 2(2)2 + (3)2 = 2(4) + 9 = 8 + 9 = 17
                                             2(2) − 3  1
                                                      = .
                                             3 + 3(2)  9

  • Making substitutions in formulae allows us to calculate the value of one quantity once the
    values of all the others are known. For example, the formula

                                                  s = 4.9t2

    gives the distance s (in meters) which an object falls in t seconds when it is dropped. So after
    t = 1 second, the object has fallen

                                       s = 4.9(1)2 = 4.9 metres.

    After t = 4 seconds, the object has fallen

                                 s = 4.9(4)2 = 4.9(16) = 78.4 metres.

  • Consider the formula
                                           f = 3x2 − 4x + 6.
    We frequently use the notation
                                         f (x) = 3x2 − 4x + 6
    to emphasise that the formula is telling us how the value of f depends upon the value of x.
    We might also write T (P, Q) = P Q in the earlier example. If we are using this notation,

                                                   4
     then f (2) means “the value of the formula when we substitute in x = 2”. So for the example
     f (x) = 3x2 − 4x + 6 we have
                               f (2)   =   3(22 ) − 4(2) + 6 = 12 − 8 + 6 = 10,
                               f (1)   =   5,
                             f (−1)    =   13,
                             f (−2)    =   26.
Exercise 2
  1. Evaluate the following expressions when x = 3, y = −2, z = 1.
      (a) x+y
      (b) 2x + 3yz
      (c) z−y
      (d) x2 + y 2 − z 2
            2x
      (e)
          y + 4z
          3xy − 2z 2
      (f)
             4xyz
      (g) x2 − 4yz.
  2. Find the value of the unknown quantity in the given formula.
      (a) F = 1.8C + 32       Find F when C = 36.
               RT
      (b) P =            Find P when R = 4, T = 275 and V = 22.
                V
      (c) s = ut + 1 at2
                   2
                              Find s when a = 9.8, u = 3.2 and t = 4.
  3. Given the formula f (x) = −2x2 + 3x + 6, find
      (a)   f (1)
      (b)   f (−1)
      (c)   f (0)
      (d)   f (2)
      (e)   f (−2).

Simplifying expressions
If you were counting the dots in Figure 1 (p.6), chances are you wouldn’t say that there are 4 dots
and 2 dots and 3 dots. You would probably say there are 9 dots. Similarly, in an expression like
                                       2x + 4y 2 + 6x − 3x + 3y 2
it make more sense to put all the same objects together. So the expression is better written as
                                               5x + 7y 2 .
When simplifying expressions in this way it is crucial to put only the same objects together.

                                                   5
                                ••
                                 •
                                 •



                                                        •
                                                       •




                                 •••




                                        Figure 1: Some dots

Multiplying brackets
We know that ab means a × b. Similarly, a(2b − c) means

                                       a × (2b − c) = 2ab − ac.

When two bracketed terms are multiplied, it means that all terms of the first bracket are multiplied
by all terms of the second. So

                              (x + 2)(x − 3) = x(x − 3) + 2(x − 3)
                                             = x2 − 3x + 2x − 6
                                             = x2 − x − 6.

and

                           (a − 3)(b + 17) = a(b + 17) + (−3)(b + 17)
                                           = a(b + 17) − 3(b + 17)
                                           = ab + 17a − 3b − 51.

After multiplying out brackets in this way, we always collect like terms together.




                                                  6
Exercise 3
  1. Simplify the following expressions by collecting together all like terms.

      (a) −4x + 2y + 6x − 18y
      (b) 2(−3x + 5y) − 4(x + 3y)
       (c) 2a2 b + 3a2 b2 − ab2 + 8a2 b.
      (d) −3(a − b) + 3(−a + b).

  2. In each case, multiply out the brackets and simplify the resulting expressions by collecting
     together like terms.

      (a) (x + 4)(x + 1)
      (b) (3y − 4)(y + 1)
       (c) (2x − y)(3x − 5y)
      (d) (2a + b − 3)(a + b)

Manipulating formulae
The formula for converting temperature in celsius (C) to temperature in Farenheit (F ) is

                                           F = 1.8C + 32.

For example, this allows us to convert 15o Celsius to 59o Farenheit. But what if we wanted to convert
from Farenheit to Celsius? Our present formula tells us immediately the value of F in terms of the
value of C. We call it an explicit formula for F . We also say that F is the subject of the formula.
What we want to do next is make C the subject of the formula, i.e. get C on its own on one side of
the formula. We must manipulate the formula to do this. There is only one rule to follow:

           Whatever we do to one side of the equation, we must also do to the other side.

The steps required to do this in the present case are as follows:

                                 F = 1.8C + 32     start here
                           F − 32 = 1.8C      subtract 32 from both sides
                           F − 32
                                   = C     divide both sides by 1.8.
                             1.8
This completes the process. We now have our Farenheit-to-Celsius formula:
                                                 F − 32
                                            C=          .
                                                   1.8




                                                  7
Manipulating formulae - examples
The rule written down above is the only rule about manipulating formulae. After that, it is a matter
of practise. There are some useful guidelines:

   • If we want to make x the subject of a formula, try to get all the terms involving x together on
     one side, with anything not involving x on the other side.

   • If x appears in the denominator of a fraction, multiply both sides of the formula by that
     denominator.

   • If there is a square root in the formula, isolate it and then square both sides.

  1. Make x the subject of the formula
                                              y = −4x − 12.
     Solution:

                                     y = −4x − 12     start here
                                y + 12 = −4x     add 12 to both sides
                                y + 12
                                       = x    divide both sides by -4
                                  −4

  2. Make Q the subject of the formula T = P Q.
     Solution:

                                   T = PQ          start here
                                   T
                                     = Q         divide both sides by P
                                   P
     There was only one step here. Dividing by P is just like dividing by any other number.

  3. Make a the subject of the formula v = u + at.
     Solution:

                                   v = u + at     start here
                               v − u = at     subtract u from both sides
                               v−u
                                     = a     divide both sides by t
                                 t




                                                  8
  4. Make x the subject of the formula
                                                         2x + 3
                                                   y=
                                                         x−1
    Solution:
                       2x + 3
                y =               start here
                        x−1
        y(x − 1)   =   2x + 3     multiply both sides by x − 1
          yx − y   =   2x + 3     expand
     yx − y − 2x   =   3    subtract 2x from both sides to get all x’s on the left
         yx − 2x   =   3+y      add y to both sides to get everthing not involving x on the right
        (y − 2)x   =   3+y      collect terms
                       3+y
                x =              divide both sides by y − 2 to finish.
                       y−2

  5. Make b the subject of
                                                            c
                                                   a=2        .
                                                            b
    Solution:
                                            c
                              a = 2               start here
                                            b
                                     c
                             a2 = 4             square both sides
                                     b
                             a2 b = 4c          multiply both sides by b
                                    4c
                                b = 2           divide both sides by a2 to finish.
                                    a

Exercise 4
  1. Make Q the subject of P = −3Q + 35.

  2. Make t the subject of v = u + at.
                                 2x + 1
  3. Make x the subject of y =          .
                                  x+4
                                 3b − 1
  4. Make b the subject of a =          .
                                 2b + 5
                                     s
  5. Make s the subject of t = 3        .
                                    s+1




                                                     9
                                  Solutions to exercises

Exercise 1
  1. (a) -4
     (b) 3bc
     (c) -32.
                                                                             d
  2. (a) Let s=average speed, d=distance travelled, t=time taken. Then s =     .
                                                                             t
     (b) Let A=area, x=length, y=breadth. Then A = xy.
     (c) Let C= temperature in Celsius, F = temperature in Farenheit. Then F = 1.8C + 32.

Exercise 2
  1. (a) 1
     (b) 0
     (c) 3
     (d) 12
     (e) 3
           5
     (f)   6
           √
     (g)    17.

  2. (a) 96.8
     (b) 50
     (c) 91.2

  3. (a) 7
     (b) 1
     (c) 6
     (d) 4
     (e) -8

Exercise 3
  1. (a) 2x − 16y
     (b) −10x − 2y
     (c) 10a2 b + 3a2 b2 − ab2
     (d) −6a + 6b



                                             10
  2. (a) x2 + 5x + 4
     (b) 3y 2 − y − 4
     (c) 6x2 − 13xy + 5y 2
     (d) 2a2 + 3ab + b2 − 3a − 3b.

Exercise 4
         35 − P
  1. Q =
            3
           v−u
  2. t =
            a
           1 − 4y
  3. x =
            y−2
           1 + 5a
  4. b =
           3 − 2a
             t2
  5. s =          .
           9 − t2




                                     11

								
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