Algorithm for solving a system of exponential equations. Exponential equations. More complex cases. Homework check
Methods for solving systems of equations
To begin with, let's briefly recall what methods of solving systems of equations exist in general.
Exist four main ways solving systems of equations:
Substitution method: any of these equations is taken and $ y $ is expressed through $ x $, then $ y $ is substituted into the equation of the system, from where the variable $ x is found. $ After that we can easily calculate the variable $ y. $
Addition method: in this method it is necessary to multiply one or both equations by such numbers so that when both are added together, one of the variables “disappears”.
Graphical method: both equations of the system are displayed on the coordinate plane and the point of their intersection is found.
Method of introducing new variables: in this method we replace any expressions to simplify the system, and then we apply one of the above methods.
Systems of exponential equations
Definition 1
Systems of equations consisting of exponential equations are called a system of exponential equations.
We will consider the solution of systems of exponential equations by examples.
Example 1
Solve system of equations
Picture 1.
Decision.
We will use the first method to solve this system. First, let's express $ y $ in terms of $ x $ in the first equation.
Figure 2.
Substitute $ y $ in the second equation:
\\ \\ \\ [- 2-x \u003d 2 \\] \\ \\
Answer: $(-4,6)$.
Example 2
Solve system of equations
Figure 3.
Decision.
This system is equivalent to the system
Figure 4.
Let's apply the fourth method for solving equations. Let $ 2 ^ x \u003d u \\ (u\u003e 0) $, and $ 3 ^ y \u003d v \\ (v\u003e 0) $, we get:
Figure 5.
Let's solve the resulting system by the addition method. Let's add the equations:
\ \
Then from the second equation, we get that
Returning to the replacement, I got a new system of exponential equations:
Figure 6.
We get:
Figure 7.
Answer: $(0,1)$.
Systems of exponential inequalities
Definition 2
Systems of inequalities consisting of exponential equations are called a system of exponential inequalities.
We will consider the solution of systems of exponential inequalities by 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, recall the following theorem on the equivalence of exponential inequalities:
Theorem 1. The inequality $ a ^ (f (x))\u003e a ^ (\\ varphi (x)) $, where $ a\u003e 0, a \\ ne 1 $ is equivalent to the collection of two systems
\ \ \
Answer: $(-4,6)$.
Example 2
Solve system of equations
Figure 3.
Decision.
This system is equivalent to the system
Figure 4.
Let's apply the fourth method for solving equations. Let $ 2 ^ x \u003d u \\ (u\u003e 0) $, and $ 3 ^ y \u003d v \\ (v\u003e 0) $, we get:
Figure 5.
Let's solve the resulting system by the addition method. Let's add the equations:
\ \
Then from the second equation, we get that
Returning to the replacement, I got a new system of exponential equations:
Figure 6.
We get:
Figure 7.
Answer: $(0,1)$.
Systems of exponential inequalities
Definition 2
Systems of inequalities consisting of exponential equations are called a system of exponential inequalities.
We will consider the solution of systems of exponential inequalities by 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, recall the following theorem on the equivalence of exponential inequalities:
Theorem 1. The inequality $ a ^ (f (x))\u003e a ^ (\\ varphi (x)) $, where $ a\u003e 0, a \\ ne 1 $ is equivalent to the collection of two systems
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