How to solve systems of exponential equations and inequalities. Exponential equations
Methods for solving systems of equations
To begin with, let us briefly recall what methods generally exist for solving systems of equations.
Exist four main ways solutions to systems of equations:
Substitution method: take any of the given equations and express $y$ in terms of $x$, then $y$ is substituted into the system equation, from where the variable $x.$ is found. After this, we can easily calculate the variable $y.$
Addition method: In this method, you need to multiply one or both equations by such numbers that when you add both together, one of the variables “disappears.”
Graphical method: both equations of the system are depicted on the coordinate plane and the point of their intersection is found.
Method of introducing new variables: in this method we replace some expressions to simplify the system, and then use one of the above methods.
Systems of exponential equations
Definition 1
Systems of equations consisting of exponential equations are called systems of exponential equations.
We will consider solving systems of exponential equations using examples.
Example 1
Solve system of equations
Picture 1.
Solution.
We will use the first method to solve this system. First, let's express $y$ in the first equation in terms of $x$.
Figure 2.
Let's substitute $y$ into the second equation:
\ \ \[-2-x=2\] \ \
Answer: $(-4,6)$.
Example 2
Solve system of equations
Figure 3.
Solution.
This system is equivalent to the system
Figure 4.
Let us apply the fourth method of solving equations. Let $2^x=u\ (u >0)$, and $3^y=v\ (v >0)$, we get:
Figure 5.
Let us solve the resulting system using the addition method. Let's add up the equations:
\ \
Then from the second equation, we get that
Returning to the replacement, I received 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 systems of exponential inequalities.
We will consider solving systems of exponential inequalities using examples.
Example 3
Solve the system of inequalities
Figure 8.
Solution:
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)) >a^(\varphi (x)) $, where $a >0,a\ne 1$ is equivalent to the collection of two systems
\ \ \
Answer: $(-4,6)$.
Example 2
Solve system of equations
Figure 3.
Solution.
This system is equivalent to the system
Figure 4.
Let us apply the fourth method of solving equations. Let $2^x=u\ (u >0)$, and $3^y=v\ (v >0)$, we get:
Figure 5.
Let us solve the resulting system using the addition method. Let's add up the equations:
\ \
Then from the second equation, we get that
Returning to the replacement, I received 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 systems of exponential inequalities.
We will consider solving systems of exponential inequalities using examples.
Example 3
Solve the system of inequalities
Figure 8.
Solution:
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)) >a^(\varphi (x)) $, where $a >0,a\ne 1$ is equivalent to the collection of two systems
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