Systems with nonlinear equations. System of equations. Detailed theory with examples (2019)

Let us analyze two types of solutions to systems of equations:

1. Solving the system using the substitution method.
2. Solving the system by term-by-term addition (subtraction) of the system equations.

In order to solve the system of equations by substitution method you need to follow a simple algorithm:
1. Express. From any equation we express one variable.
2. Substitute. We substitute the resulting value into another equation instead of the expressed variable.
3. Solve the resulting equation with one variable. We find a solution to the system.

To solve system by term-by-term addition (subtraction) method need to:
1. Select a variable for which we will make identical coefficients.
2. We add or subtract equations, resulting in an equation with one variable.
3. Solve the resulting linear equation. We find a solution to the system.

The solution to the system is the intersection points of the function graphs.

Let us consider in detail the solution of systems using examples.

Example #1:

Let's solve by substitution method

Solving a system of equations using the substitution method

2x+5y=1 (1 equation)
x-10y=3 (2nd equation)

1. Express
It can be seen that in the second equation there is a variable x with a coefficient of 1, which means that it is easiest to express the variable x from the second equation.
x=3+10y

2.After we have expressed it, we substitute 3+10y into the first equation instead of the variable x.
2(3+10y)+5y=1

3. Solve the resulting equation with one variable.
2(3+10y)+5y=1 (open the brackets)
6+20y+5y=1
25y=1-6
25y=-5 |: (25)
y=-5:25
y=-0.2

The solution to the equation system is the intersection points of the graphs, therefore we need to find x and y, because the intersection point consists of x and y. Let's find x, in the first point where we expressed it we substitute y.
x=3+10y
x=3+10*(-0.2)=1

It is customary to write points in the first place we write the variable x, and in the second place the variable y.
Answer: (1; -0.2)

Example #2:

Let's solve using the term-by-term addition (subtraction) method.

Solving a system of equations using the addition method

3x-2y=1 (1 equation)
2x-3y=-10 (2nd equation)

1. We choose a variable, let’s say we choose x. In the first equation, the variable x has a coefficient of 3, in the second - 2. We need to make the coefficients the same, for this we have the right to multiply the equations or divide by any number. We multiply the first equation by 2, and the second by 3 and get a total coefficient of 6.

3x-2y=1 |*2
6x-4y=2

2x-3y=-10 |*3
6x-9y=-30

2. Subtract the second from the first equation to get rid of the variable x. Solve the linear equation.
__6x-4y=2

5y=32 | :5
y=6.4

3. Find x. We substitute the found y into any of the equations, let’s say into the first equation.
3x-2y=1
3x-2*6.4=1
3x-12.8=1
3x=1+12.8
3x=13.8 |:3
x=4.6

The intersection point will be x=4.6; y=6.4
Answer: (4.6; 6.4)

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The use of equations is widespread in our lives. They are used in many calculations, construction of structures and even sports. Man used equations in ancient times, and since then their use has only increased. Surely many people know that the equation is a kind of identity with an unknown, which must be determined in order to solve the equation and obtain equal values ​​of the left and right sides. To solve of this kind equation, it is necessary to transfer all known values ​​to the left side, and all unknown values ​​to the right side. These equations can be solved using 3 methods:

1) substitutions;

2) addition;

3) plotting.

The choice of method depends on the target equation. Decide online equation with two unknowns it is possible on many sites, but you should not blindly trust the result obtained.

Below is an example of solving an equation with 2 unknowns using the addition method.

\[-9x + 5y = -40\]

The first thing to start solving with is to add each term taking into account their signs:

\[-5y + 5y = 0\]

In most cases, one of the sums that includes the unknown will contain a value equal to zero. At the next stage of solving the equation, we need to create an equation from the data obtained:

\[-7x + 0 = 21\]

Find the unknown:

\[-7x = 21, x = 21 \div (-7) = -3\]

Insert the resulting value into any of the original equations and obtain the 2 unknown by solving the equation linear type:

\[-6 - 5y = 61\]

\[-5y = 61 + 6\]

Final result:

Where can I solve an equation with 2 unknowns online?

You can solve an equation with two unknowns using an online solver on the website https://site. The free online solver will allow you to solve online equations of any complexity in a matter of seconds. All you need to do is simply enter your data into the solver. You can also watch video instructions and learn how to solve the equation on our website. And if you still have questions, you can ask them in our VKontakte group http://vk.com/pocketteacher. Join our group, we are always happy to help you.

Chapter 8. Systems of equations

8.2. System of two linear equations with two unknowns

Definition

Several equations in which the same unknowns denote the same quantity are called system of equations.
The type system is called normal form systems of two linear equations with two unknowns.
Solving such a system means finding the set of all solutions common to both equations.

How to solve such a system?

Such a system can be solved, for example, graphically. Typically, such a system is graphically represented by two straight lines, and the general solution to these equations (the solution to the system) will be the coordinates of the common point of the two straight lines. There are three possible cases here:
1) Straight lines (graphs) have only one common point (intersect) - the system of equations has a unique solution and it is called definite.
2) Straight lines (graphs) do not have common points (parallel) - the system has no solution and it is called inconsistent.
3) Straight lines (graphs) have infinitely many common points (coincide) - the system has an infinite number of solutions and is called indefinite.

There's something I don't understand yet. Maybe it will be clearer with examples?

Of course, now we will give an example for each case and everything will immediately become clearer.

Let's start with an example when the system is defined (has a unique solution). Let's take the system. Let's build graphs of these functions.

They intersect only at one point, therefore the solution to this system is only the coordinates of the point: , .

Now let's give an example of an incompatible system (one that has no solution). Let's consider such a system.

In this case, the system is contradictory: the left parts are equal, but the right parts are different. The graphs do not have common points (parallel), therefore the system has no solution.

Well, now there is the last case, when the system is uncertain (has an infinite number of solutions). Here is an example of such a system: . Let's plot these equations.

Straight lines (graphs) have infinitely many common points (coincide), which means the system has an infinite number of solutions. In this case, the equations of the system are equivalent (multiplying the second equation by 2 , we get the first equation).

The most important is the first case. The only solution to such a system can always be found graphically - sometimes exactly, and most often approximately with the required degree of accuracy.

Definition

Two systems of equations are called equivalent (equivalent), if all solutions of each of them are also solutions of the other (the sets of solutions coincide) or if both have no solutions.

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