What is an equation

An equation is one of the fundamental concepts in mathematics and one of the means by which all mathematics works.

Introductory example

An equation has a left side, then an equals sign and a right side. A trivial equation might look like this:

$$x=2$$

On the left side is the variable x, followed by the equals sign, and on the right side is the number 2. This equation is simple and tells us that the value of the variable x is equal to two. The variable x then usually represents something we are looking for. For example, it could be the number of buttons on a shirt or an employee's salary.

However, an equation in this form is something we are looking for rather than something that would be in the example specification. In practice we have some more complicated equation, for example we are told how much five employees earn in total and our task is to find out how much one employee earns (assuming they all earn the same).

For example, we know that five teachers earn a total of 120,000 crowns per month. How much does one teacher earn? Let's set up the equation like this: we'll label the salary of one teacher as x. This gives us the equation:

$$5x=120 000$$

This equation is the mathematical notation of the assignment "five teachers have a total salary of 120 000". Our goal is to find the salary of one teacher, i.e. to find the value of x = ?. We'll jump ahead a bit and use equivalent equation adjustments and divide the equation by five. If five teachers are paid 120 000, then one teacher is paid five times less. As a result, we have

$$x=120 000 / 5$$

and after dividing by

$$x=24 000.$$

The definition of the equation

To define the concept of an equation, we will need to know what a function is. If we know functions, then we can think of an equation as writing the equality of two functions:

$$f(x)=g(x)$$

This is the general notation for an equation with one unknown. On the left hand side we have the function f and on the right hand side the function g. Our task is to find the roots of the equation, which are the values of x, for which the functions f and g have the same value.

So we want to find those particular values of x, I'll label them x₁, for which the

$$f(x_1)=g(x_1)$$

Let's take the equation 2x = −4x + 6 to help us out. Then f(x) = 2x and g(x) = −4x + 6 would hold. We are looking for such x, for which the function f has the same value as the function g. Solving the equation using equivalence adjustments, we get:

$$\begin{eqnarray} 2x&=&-4x+6\\ 2x+4x&=&-4x+4x+6\\ 6x&=&6\\ x&=&1 \end{eqnarray}$$

The equation results in the value x = 1. For this value, both functions should return the same value. If we call the function f with a one, we get

$$f(1)=2\cdot1=2$$

The function f has a value of two at point two. What about the function g? Let's call it with a one:

$$g(1)=-4\cdot1+6=-4+6=2$$

We see that we get a two again. So the number one is the root of the equation. It is also the only root of the equation in the domain of real numbers. If we call the function with a different value, we get different results. For example, for five we would get:

$$\begin{eqnarray} f(5)&=&2\cdot5=10\\ g(5)&=&-4\cdot5+6=-20+6=-14 \end{eqnarray}$$

Graphical Meaning

The solutions to the equation have a nice and obvious graphical meaning. That's because if you draw graphs of the functions that occur on the left and right sides of the equation, then those graphs will intersect at the exact points where the equation has a solution.

Let's go back to the equation 2x = −4x + 6. The function on the left side is f(x) = 2x and the function on the right side is g(x) = −4x + 6. We know that the root of this equation is the value x = 1. So it should be true that at point two these functions intersect. The following is a picture of the graphs of the two functions:

You can see that the lines intersect at one point, and this point has the coordinates [1,2]. The first coordinate is the value of x, the root of the equation. The second value is the value of y, the resultant value of both functions if you call them with a one.

Thus, the solution to the equation is all the points at which the functions on the left and right sides intersect. Often we convert equations so that the equation on the right-hand side is zero, i.e., the constant function g(x) = 0. Then we solve when the function on the left-hand side intersects the x axis. We can modify the previous equation 2x = −4x + 6 so that the right side is zero as follows:

$$\begin{eqnarray} 2x&=&-4x+6\\ 6x&=&6\\ 6x-6&=&0 \end{eqnarray}$$

If we draw the graph of the function on the left-hand side...

we find that the graph intersects the x axis at point 1. We can see that even if we modified the equation to have zero on the right-hand side, we get the same result - one again.

The number of solutions to the equation

From the graphical interpretation of equations, we can infer how many different solutions an equation can have. We can easily answer questions like - does each equation have a solution? Can an equation have more than one solution? Can an equation have infinitely many solutions?

No solutions

Let's start in order. Is there an equation that has no solution? We reduce this to a question - are there any two graphs of functions that never intersect? Of course there are. If we stick with linear equations that have straight lines as graphs, then just take lines that are parallel. For example:

$$x+1=x+2$$

Just by common sense, we can deduce that x + 1 can never simultaneously equal x + 2. If we draw the graph, we get:

We can see that the functions are lines that are parallel and so never intersect. Thus, the equation we have put together has no solution. We can modify the equation as follows:

$$\begin{eqnarray} x+1&=&x+2\\ x-x&=&2-1\\ 0x&=&1\\ 0&\ne&1 \end{eqnarray}$$

We see that the equation really has no solution.

We can think of many other equations that will have no solution, for example, the equation x² = x − 1. The graphs of the two functions never intersect:

More solutions

Can an equation have more than one solution? (But still fewer than infinity.) Again, this reduces to the question - are there any two graphs of functions such that they have a finite number of intersections and the number of intersections is also greater than two?

They certainly exist. In the previous chapter, we had an equation with a quadratic and a linear function as our last example. We can modify the previous graph by shifting the line so that it intersects the graph of the quadratic function at exactly two points. Just take the function y = x instead of y = x − 1.

We can see that the graphs intersect at two points. We could solve the equation as follows:

$$\begin{eqnarray} x^2&=&x\\ x^2-x&=&0\\ x\cdot(x-1)&=&0\\ x_1&=&0\\ x_2&=&1 \end{eqnarray}$$

Infinitely many solutions

Are there equations that have infinitely many solutions? That is, are there two graphs such that they intersect at infinitely many points? Yes, there are. Just take some periodic function and interleave it with a straight line in some convenient way. A typical periodic function is the sine. The sine has a range of values of the interval <−1, 1> and repeats periodically in that range.

So just put sin(x) = a, where a will be from the range of values. A concrete example might look like this: sin(x) = 0,5. Graphs of functions:

We can see that in the figure the line intersects the sine at several points. The sine continues to undulate on both the right and the left, so on both the right and the left the line still intersects the sine infinitely many times.

The same functions in the equation

Can there be a situation where we can substitute any value from the definitional range of functions after x and the equation will hold? This will only happen when the two functions are the same or when we can fit one function to the other. So an example:

$$2x+4=2\cdot(x+2)$$

At first glance, we have different functions on each side, but if we multiply the parenthesis on the right side, we get the same functions:

$$2x+4=2x+4$$

Such an equation has a solution set equal to the defining domain of functions, so the solution set is equal to the set of real numbers. Such an equation can be modified to the form 0 = 0.

$$\begin{eqnarray} 2x+4&=&2x+4 \qquad/-2x\\ 4&=&4 \qquad /-4\\ 0&=&0 \end{eqnarray}$$