Chapter 2 Equations in one Variable

Section 2.1 Simple Equations

2.1.1 Introduction

Info 2.1.1
 
An equation is an expression of the form

left-hand side   =   right-hand side  

with mathematical expressions on both sides of the equation. These expressions generally contain variables or unknowns (e.q.  x). Depending on the variable values an equation is satisfied if both sides of the equation evaluate to the same value. An equation is not satisfied if the sides of the equation evaluate to different values.

Equations describe relations between expressions or model a problem to be solved. In general, an equation itself is not true or false. Instead, some variables satisfy the equation and others do not. To test whether the equation is true or false for a single variable value this value has to be inserted into the equation. Then, both sides of the equation are evaluated to certain values. The equation is satisfied by an inserted variable value if the evaluated values coincide:
Example 2.1.2
The equation 2x-1= x2 has the right-hand side x2 and the left-hand side 2x-1. Inserting x=1 results in the value 1 on both sides of the equals sign, hence x=1 is a solution of this equation. However, x=2 is no solution since the left-hand side of the equation is evaluated to the value 4 while the right-hand side is evaluated to the value 3.

Info 2.1.3
 
The solution set L of an equation is the set of all numbers satisfying the relation

left-hand side   =   right-hand side  

if inserted into the the equation instead of the variable (e.q. x).

Typical problems concerning equations are:
  • specify the solution set of an equation, i.e. find all variable values satisfying the equation,
  • transform the equation, in particular, solve an equation for the variables, and
  • find an equation modelling a problem described textually.

Example 2.1.4
We want to design a savings deposit in such a way that there is a fixed annual return. The bank wants to make sure that when investing over a five-year period, a saver receives exactly 600 Euros more than when investing over a two-year period.
First, the word problem is translated into an equation with the variable r denoting the annual return. The resulting equation is 5·r=2·r+600. It says that five payments (left-hand side of the equation) equal two payments plus 600. (For simplicity, we omit the unit Euro during calculation.)
We can easily solve this equation can be solved for r by subtracting the term 2r from both sides of the equation. The resulting equation reads 3r=600, and dividing by 3 results in the solution r=200.
Thus, the bank has to offer a return of 200 Euros per year to reach the required savings target.

Info 2.1.5
 
Two equations are said to be equivalent if they have the same solution set.
An equivalent transformation is a special transformation that changes the equation but not its solution set. Important equivalent transformations are
  • adding/subtracting terms to both sides of the equation,
  • exchanging both sides of the equation,
  • transformation of terms on one side of the equation, and
  • substituting a term for another that is known to always have the same value.

The following transformations are equivalent transformations only if the used term is known to be non-zero (which may depend on the possible values of the variables):
  • multiplying/dividing by a term (this term has to be non-zero),
  • taking the reciprocal on both sides of the equation (both sides have to be non-zero).

Here, the following notation is used:
  • equivalent equations are indicated by the symbol ⇔ (which reads: if and only if, i.e. one equation is satisfied if and only if the other equation is satisfied).
  • under this symbol we put a short description of the transforming operation (or, for solutions with more than one line, the transforming operation is written next to the transformation).

What matters is that the reader should be able to understand which transformation was carried out.
Example 2.1.6
This example illustrates two simple equivalent transformations written in a single line. Even though the symbol ⇔ is two-sided the notation is interpreted in such a way that the transformation is applied from left to right:

3x- x2   =  2x- x2 +1     ⇔ ⏟ + x2     3x  =  2x+1     ⇔ ⏟ -2x     x  =  1 .

The leftmost equation and the rightmost equation are equivalent. On the left we have the initial equation (corresponding to a certain word problem) and on the right we have an equivalent equation showing the solution immediately.

Example 2.1.7
For several complicated transformations, the transformation steps should be written one under another. In this case we use vertical bars to separate the respective transformations from the equations.


Start: 12+t  =   2t 2 t2 +t    ‖                -t    ⇔ 12  =   2t 2 t2     ‖                 sides exchanged     ⇔ 2t 2 t2   =  12    ‖                 left-hand side transformed     ⇔ 1 t   =  12    ‖                 reciprocals taken     ⇔ t  =   1 12  .


Here, after the vertical bar both short symbols as, e.g. -t, and textual descriptions are allowed. Again, what matters is that the reader can understand which transformations were carried out and so can check that everything was done correctly.

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