Systems of inequalities, solving systems of linear inequalities

The concept of the system and its rozvytku

Definition: If the task is to find all common solutions of two (or more) inequalities on the one or more variables, then we say that it is necessary razvesti system of inequalities.

Definition: Rozvyazka system — the value of a variable or such an ordered set of values smya that satisfies all the inequalities in the system, that is rozvyazka system of two or more inequalities with unknown call such an ordered set a lot of numbers, after the substitution of which into the system is unknown, all inequalities turn into true numerical equality.

Definition: Razvesti the system of equations is to find all its interchanges , or to prove that they are not.

If the system has no solution, then it is incompatible.

Example systems of inequalities

the system of three equations with two variables

Couple that is one of rozvytku system

Scheme the solution of systems of inequalities with one variable

  1. Rozvadov each inequality separately.
  2. To find all joint junction of these inequalities.

Scheme the solution of systems of inequalities with multiple variables

  1. Rozvadov the system of inequalities as a system of equations by changing, for a while, the inequality sign to an equal sign.
  2. To change the sign back and to find common solutions these inequities.

Examples of solving systems of equations

Solution by graphical methods

Example 1

Rozwarte equation:


Building graphics

Building graphics will see that the graphs intersect at the point


The decision of the method of substitution

Example 2

Rozwarte equation:


From the first equation we Express And substitute the resulting expression into the second equation of the system:

The resulting value is substituted into the expression


Solution by adding

Example 3

Rozwarte equation:


Need to get rid of a variable poslano Multiply the first equation of system 3 and the second by 2.

Add pocino equation and get:

Find the value of the first equation of the system:


Remark: the method add can be multiplied not only on positive numbers, and negative.

How rozwiazywanie the system of equations to solve only to You.

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