The lesson of the system of indicative equations and inequalities. Indicative equations
Methods for solving systems of equations
To begin with, we briefly remember which ways there are ways to solve systems of equations.
Exist four basic ways solutions of systems of equations:
Method of substitution: Any of these equations is taken and $ y $ is expressed by $ x $, then $ y $ is substituted into the system equation, from where there is a $ x variable. $ After that, we can easily calculate the $ y variable. $
Method of addition: In this method, it is necessary to multiply one or both equations for such numbers so that when adding together both, one of the variables "disappeared".
Graphic method: Both system equations are depicted on the coordinate plane and the point of their intersection is located.
The method of introducing new variables: In this method, we make the replacement of any expressions to simplify the system, and then apply one of the above methods.
Systems of indicative equations
Definition 1.
Systems of equations consisting of indicatory equations, are called the system of indicative equations.
Solving systems of indicative equations will be considered on the examples.
Example 1.
Solve the system of equations
Picture 1.
Decision.
We will use the first way to solve this system. To begin with, express in the first equation $ y $ one $ x $.
Figure 2.
Substitute $ y $ to the second equation:
\\ \\ \\ [- 2-x \u003d 2 \\] \\ \\
Answer: $(-4,6)$.
Example 2.
Solve the system of equations
Figure 3.
Decision.
This system is equivalent to the system.
Figure 4.
Apply the fourth method of solving equations. Let $ 2 ^ x \u003d u \\ (u\u003e 0) $, and $ 3 ^ y \u003d V \\ (V\u003e 0) $, we get:
Figure 5.
We solve the resulting system by the method of addition. Mixion equations:
\ \
Then from the second equation, we get that
Returning to replacement, received a new system of indicative equations:
Figure 6.
We get:
Figure 7.
Answer: $(0,1)$.
Systems of indicative inequalities
Definition 2.
Inequality systems consisting of indicative equations are called the system indicative inequalities.
Solving systems of indicative inequalities will be considered on the examples.
Example 3.
Solve the system of inequalities
Figure 8.
Decision:
This system of inequalities is equivalent to the system
Figure 9.
To solve the first inequality, we will recall the following equivalence theorem of indicative inequalities:
Theorem 1. Inequality $ a ^ (f (x))\u003e a ^ (\\ varphi (x)) $, where $ a\u003e 0, a \\ ne $ 1 is equivalent to the combination of two systems
\ \ \
Answer: $(-4,6)$.
Example 2.
Solve the system of equations
Figure 3.
Decision.
This system is equivalent to the system.
Figure 4.
Apply the fourth method of solving equations. Let $ 2 ^ x \u003d u \\ (u\u003e 0) $, and $ 3 ^ y \u003d V \\ (V\u003e 0) $, we get:
Figure 5.
We solve the resulting system by the method of addition. Mixion equations:
\ \
Then from the second equation, we get that
Returning to replacement, received a new system of indicative equations:
Figure 6.
We get:
Figure 7.
Answer: $(0,1)$.
Systems of indicative inequalities
Definition 2.
Inequality systems consisting of indicative equations are called a system of indicative inequalities.
Solving systems of indicative inequalities will be considered on the examples.
Example 3.
Solve the system of inequalities
Figure 8.
Decision:
This system of inequalities is equivalent to the system
Figure 9.
To solve the first inequality, we will recall the following equivalence theorem of indicative inequalities:
Theorem 1. Inequality $ a ^ (f (x))\u003e a ^ (\\ varphi (x)) $, where $ a\u003e 0, a \\ ne $ 1 is equivalent to the combination of two systems
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