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How to solve systems of equations?

The system of equations is, at first glance, an absolutely useless and unnecessary thing in life. But if you look at the essence of the processes around us, in nature, in the achievements of science and technology, it becomes clear - this is not so. Almost any phenomenon can be described by a system of equations, starting from rain in spring, ending with the flight of asteroids in space. And, as is well known, a phenomenon for which a sufficiently accurate description is determined can be predicted.

What is a system of equations

A system is a number of ordinary equations that must be executed simultaneously. On the letter, the system is indicated by a curly bracket on the left side, uniting all the equations. How to solve a system of equations? Due to the fact that all the equations of each given system must be in force together, several ways are opened to alter, transform the system without changing its roots. Such transformations are called equivalent. For example, there is a system “x + y = 2; x - y = 0 "Obviously, its roots are" x = 1; y = 1 ". Consider equivalent transformations.


Solving a system of equations by the method of addition is easiest. Let us add the second to the first equation, and completely the left and right sides. Get the system "2 * x + 0 * y = 2 + 0; x - y = 0 ". In the first equation of the system we find the root x = 1. We substitute it into the second equation and get the value of the second variable y = 1. The system is solved. It should be remembered that before adding the equation can be fully multiplied by a constant, which is also an equivalent transformation. And this constant does not have to be positive.


If the task sounds like “solve the system of equations by the substitution method” - everything is somewhat worse. The substitution method is more horzdozdok in comparison with the addition method and more than one tetrad sheet can be spent on one small system. In order to solve a system of equations by the substitution method, we must take one of the equations (for convenience, the first one) and express one of the variables from it (for convenience, the first one again). We get an equation like "x = 2y + 7z + 9a -2b - 11" Now, in all other equations, instead of x, we substitute the expression "2y + 7z + 9a -2b - 11" chosen for it, remembering to multiply it by original equation. We obtain an equation expressing x and several equations that are independent of x.Then we perform a similar operation for all variables. In the last equation will be clearly expressed last variable, this is a feature of this method. And, as a result, it can be found. Knowing the last variable, you can substitute its numerical value in the penultimate equation and find the penultimate variable. Continuing such frauds, you can find all the variables. When solving tasks for such a method, one should be very careful - it is difficult not to get confused in this mass of arithmetic operations and not to make an annoying mistake. To verify the correctness of the solution, you can use the service. This is a powerful computing center that can answer a wide variety of requests; you just have to describe the task so that the computer understands it.


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