mozaMedical by Mozaik Education

The concept of equation, identity

In our everyday life or when examining the laws of nature we often come across such questions which are not easy to answer. The reason for it is primarily that our understanding can calculate several steps in advance only to a limited extent. Therefore the reasonable people looked for such tools with the help of which they could improve this capability.

One of these is a quite productive tool of mathematics, the equation.

Example 1

Let us give the number for which the following is true: if we take away 1 from its double, then the result is equal to the value resulting when taking away three times the number from 2.

Solution

Let the missing number be x. It can be quite efficient later on too if we first answer the question formulated in the exercise (naturally in the way that we denote the unknown number by some letter).
Then we express the properties formulated in the exercise using the language of mathematics:

\latex{2x-1=2-3x.}

Thus we get an equation, which we keep transforming until we get the value of the unknown from it. We can easily realise that only \latex{x=\frac{3}{5}} satisfies our condition in this case.
This value is called the solution of the equation or the root of the equation.

Example 2

Let us solve the following equation on the set of integers.

\latex{2(x+3)-4=2x+2.}

Solution

\latex{2x+6-4=2x+2},

\latex{2x+2=2x+2}.

After rearranging the equation finally we get an equation, which will be true in all cases irrespectively of the value we give to variable x. Therefore every \latex{x\in\Z} is a solution. This type of equation is called an identity.

⯁ ⯁ ⯁

There is a fundamental set belonging to each equation, on which we are looking for the solutions. So for example in example 2 this set is the set of integers. The domain of the equation is the largest subset of the fundamental set on which the expressions appearing in the equation have a meaning. If we do not give the fundamental set in advance, then the fundamental set is the set of real numbers.

Based on our knowledge we can give even two definitions for the concept of equations:

With the help of the equation we actually formulate a declarative sentence in which one or maybe more unknowns, variables can be found.

The sentences, about which it can unambiguously be decided whether they are true or false, are called propositions or statements in mathematical logic. Every single statement has a logic value of either true or false.

Based on the above it can be determined what we mean by equation, we can give its concept. Equations can be considered as logic functions, i.e. as incomplete statements the logical values of which depend on what we substitute for the variable(s). For example the statement of equation \latex{2x-1=2-3x} takes the true or false value depending on whether we substitute the value \latex{x=\frac{3}{5}} or something else for \latex{x}. While solving the equation we always try for finding such values x for which the logical value of the statement is true.

We can give a more graphical definition for the concept of the equation in which we can use the concepts learnt for the functions.

It can be applied if we think of the equations as the expressions on each side of the equalities would be the assignment rules of separate functions. So we can find the solution of example 1 by plotting the graphs of the functions on the two sides in a common coordinate system. (Figure 1)

While solving the equation we are looking for such values x of the fundamental set for which the substitution values of the two functions are equal, i.e. \latex{f(x) = g(x).} (Figure 2)

Figure 2

\latex{ y }

\latex{ x }

\latex{ f }

\latex{ g }

Exercises

{{exercise_number}}. Which of the following sentences can be considered as statements, and which are true and which are false?

Today is Friday.
The two diagonals of a parallelogram divide the parallelogram into four parts with equal areas.
Number 2 is the one and only even prime number.
Snow White is more beautiful than her wicked stepmother.
The square of every even number is divisible by \latex{8}.
If we add \latex{15} to x, then we get \latex{18} as the result.
It is going to rain tomorrow.

{{exercise_number}}. What can we write in the place of the variables in the following logic functions so that we get true and what to get false statements?

All the angles of quadrilateral x are of equal measure.
Number c is the smallest natural number.
If a number is divisible by x, then it is also divisible by 12.
If a number is divisible by 12, then it is also divisible by y.
If we take away 3 from the double of x, then we get 15.
Number n is a whole number which is at least –2 and at most 4.

{{exercise_number}}. Give equations which, after being solved, give answer to the questions formulated in parts a)–e).

For which number is it true that it is two greater than its double?
For which number is it true that it is three less than its triple?
We added ten to a number, then we multiplied it by two, thus we got a number three times the original number. What was the original number?
Which number did we think of if the number 7 less than its triple is 5 greater than its double?
We multiplied a number by 6, then we added 6 to it, thus we got the one-sixth of a number 6 less than 258. What was the original number?

{{exercise_number}}. Give the largest subset of the set of real numbers on which we can look for the solutions of the following equations.

\latex{\frac{1+x}{x-2}=7;}

\latex{\frac{1+x}{2-x}+\frac{2-x}{1+x}=2;}

\latex{\frac{(x-1)^2}{x(x-2)}=1;}

\latex{\frac{x^3}{x(x^2-1)}=7;}

\latex{\frac{4x-3}{3x(12x-9)}=\frac{3x(21x-9)}{4x-3};}

\latex{\frac{2}{x-1}+\frac{3}{x+1}=\frac{5}{x^2-1};}

\latex{\frac{1}{x^2-2x+1}-\frac{1}{x^2+2x+1}=\frac{2}{x^2-1};}

\latex{\frac{1}{25x^2-15x}-\frac{1}{20x-12}=\frac{1}{30x-18}}.

{{exercise_number}}. In the following equations a denotes a real parameter. Can we substitute value a so that the given equations become identities? Write such values for the place of the parameter for which the given equations do not have a solution.

\latex{3x-4=ax-4}

\latex{7(x-2)=7x+a}

\latex{-3-4x=a\left(x+\frac{3}{4}\right)}

\latex{2(2x-3)+3(2-x)=ax}

{{exercise_number}}. Consider the functions on the two sides of the following equations, and then by plotting their graphs find the points at which they intersect.

\latex{-x+4=3x}

\latex{2x-1=-x+2}

\latex{\left|x\right|=2x-3}

Quiz

The below five statements can be read on a piece of paper. Which of the statements are true and which are false?

there is

false statements on this paper

there are

1

2

3

4

5

Mathematics 9.

-