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Variables and Terms
Example
To be accepted by Swiss Post as a bulky good, a parcel has to fulfill the following format conditions :
$2 \times width + 2 \times height + length \le 400\;cm$
In this description Swiss Post uses verbal terms as width, height, length. They serve as placeholders for values that are not known at the moment or aren't intended to be specified.
In Mathematics these plaeceholders for quantities are referred to as variables; conventionally, they are labelled by single letters (instead of words).
In our example:
w = width
h = height $\quad \quad \quad$
Restriction: $ \; 2 \cdot w + 2 \cdot h + l \le 400$
$ l $ =
length
Expressions consisting of numbers, variables and operators (like $ \; 2 \cdot w + 2 \cdot h + l $ in the example) are also referred to as terms.
Formulas
Example:
The tickets for a sports event are sold at 10 CHF for adults and at 4 CHF for children. There are fixed costs (playground hire, referee) of 400 CHF. What ist the total profit if there are 400 adults and 150 children paying?
This question shouldn't be too difficult to answer; the total profit can be calculated as follows:
\(400 \cdot 10 + 150 \cdot 4 - 400 = 4'200\)
You get the revenue by multiplying the number of adults by 10, the number of children by 4, and then addding both. Finally you have to subtract the fixed cost.
However, the "role" of the figure 400 in the calculation is not quite clear without extra explanation.
Thus, it makes sense to introduce variables:
x: number of adults
y: number of children
f: fixed costs (in CHF)
g: total profit (in CHF)
With these variables we can write down the profit formula:
\(g = 10 \cdot x + 4 \cdot y - f\)
or shorter
\(g = 10x + 4y - f\)
It allows to calculate the total profit g for any combinations of the "input values" x, y und f.
To solve a computational problem like the one shown above, it is essential to specify the "solution process". Variables allow us to do this in a very simple way. The resulting formula can be used for any input figures, and moreover, the computing work can be transferred to a pocket calculator or a computer.
In the previous example ("bulky parcel") the critical overall measure t can be described by the following formula:
\(t = 2b + 2h + l\)
Rearranging terms
In many applications you will need to rearrange or simplify algebraic terms. This must be done without changing the meaning of these terms. Two terms are said to be equivalent if they produce the same number for any possible value of the variable(s).
For example:
\( t = 2b + 2h + l \)
can be transformed to
\(t = 2(b + h)
+ l \)
AnExcel- Sheet allows you to check the equivalence of the two terms for some specific input values.
Please note that checking several cases doesn't prove the equivalence in a stringent way. But comparing the terms with different values can help you to discover incorrect transformations.
Example:
Is \( {(a + b)^2} = {a^2} + {b^2}\) ?
No! Since \({(3 + 2)^2}
= {5^2} = 25\) , but \({3^2} + {2^2} = 9 + 4 = 13\) , the equation
above cannot be true.
The (incorrect) transformation above is a "classical" error in manipulating algebraic expressions.
For doing better, see the topic Binomial Formulas of this tutorial.
In order to prove the correctness of a transformation you have to refer to general algebraic rules. The transformation above, for example, is an application of the Distributive Law:
\(a(b + c) = ab + ac\)
Here you'll findsome other important rules at a glance.
Many of the following pages of this tutorial deal with the correct application of these rules. You can get an overview by opening the side bar.
Algebraic Structure
To rearrange algebraic terms correctly, it is important to recognise its overall structure.
In our example:
 |
is a sum to start with; the two summands for their part are products. |
 |
is a product; the second factor is itself a sum. |
The "structure boxes" you see in this example are a tool to describe the structure of an algebraic term. We might also use verbal descriptions such as "has the form \(A+B\) " (in the first case) or "has the form \(A \cdot B\)" (in the second case).
Example:
Which of the following expressions have the form \(A \cdot B + C
\) ?
a) \(2x + 3y\)
b) \(x \cdot (y + 1)\)
c) \({w^2}
- v \qquad \qquad \qquad \qquad \qquad \qquad \) Solutions
Equations
If we connect two algebraic terms by an equals sign = , we get an equation. An equation can have different meanings:
Formulas like \(2b + 2h = 2(b + h)\) are universally valid; no matter what input values you chose for the variables b and h, both sides of the equation must always be the same.
In other cases, equations describe a problem to be solved: you try to find values for the variable that "satisfy" the equation, i.e. that make it a true statement.
There are different approaches for solving an equation; see more details in the chapter Equations and Systems of Equations.
1)
Mr Smith wants to buy furniture:
1 table for a dollars, 4
chairs for b dollars each and 2 cupboards for c dollars each.
a)
Give a formula for the overall price P of Mr Smith's purchases.
b)
Mr Smith makes an advance payment of z dollars and then pays the
rest by 5 equal installments.
Give a formula for the amount r
of one installment. (Do not account for interest.)
2)
To the right you see some coins arrangend in a square. Every complete, i.e. uninterrupted lineup contains n coins, no matter if it is horizontal, vertical or diagonal.
a)
How many coins do you count in the case n = 7 (as in the figure)?
b)
Give a general formula for the number a of coins, provided there is an odd number n of coins in each lineup. How many coins are needed to place 13 coins in each lineup?
c)
Does your formula hold for even numbers n as well?
3)
Write an algebraic expression that will symbolize each of the following terms:
a)
Twice the number z.
b)
The number x is increased by 1 and the result is multiplicated by 5.
c)
2 less than three times the number a.
d)
The number n is subtracted from m and the result is increased by 3.
e)
Three quarters of p.
4)
A group of college students consists of m males and f females.
What is the meaning of the following equations?
(Give a short
verbal interpretation of the stated equality.)
a)
\( m+f=30 \)
b)
\( m=2f \)
c)
\( m+2=f \)
d)
\( \large \frac {m}{m+f} \normalsize = 0.4 \)
5)
Analyze the algebraic structure of the following terms as far as possible; make use of the "structure boxes" as introduced in the theory section:
Example: \( \quad 2x-3(y-1) \) can be represented as
a)
\( (p-1)(2p+1) \)
b)
\( a+ \large \frac {b}{c-2} \)
c)
\( r(s+2)+2t \)
6)
Which of the following expressions have the form A+B∙C ?
In
every case, indicate what terms are represented by A, B and C –
one possibility will be sufficient.
b)
\( (x+2) \cdot (y+2) \)
c)
\( x+2 \cdot (y+2) \)
d)
\( x+2 \cdot y+2 \)
More training options can be found under the following links:
MathGoodies: Writing algebraic expressions
Massey University (NZL): Nice examples for constructing formulae
The MathPage: Algebraic Expressions Theory and exercices
If you like the 'Millionaire' Game Show:
Math-Play: Milionaire Game Recognizing algebraic expressions