Math Problem Statement
Решите систему:
Solution
Let's solve the given system of equations:
5^{x+y} = 125, \\ 4^{(x-y)^2 - 1} = 1. \end{cases}$$ ### Step 1: Solve the first equation The first equation is: $$5^{x+y} = 125.$$ Since $$125 = 5^3$$, we can write: $$5^{x+y} = 5^3.$$ Equating the exponents (since the bases are the same), we get: $$x + y = 3.$$ ### Step 2: Solve the second equation The second equation is: $$4^{(x-y)^2 - 1} = 1.$$ We know that any non-zero number raised to the power of 0 is 1, so: $$(x-y)^2 - 1 = 0.$$ Solving for $$(x-y)^2$$, we get: $$(x-y)^2 = 1.$$ Taking the square root of both sides, we get two possible solutions: $$x - y = 1 \quad \text{or} \quad x - y = -1.$$ ### Step 3: Solve the system Now we have two cases to consider: **Case 1: $$x - y = 1$$** - From $$x + y = 3$$ and $$x - y = 1$$: $$\begin{cases} x + y = 3, \\ x - y = 1. \end{cases}$$ Adding these two equations: $$2x = 4 \quad \Rightarrow \quad x = 2.$$ Substituting $$x = 2$$ into $$x + y = 3$$: $$2 + y = 3 \quad \Rightarrow \quad y = 1.$$ **Case 2: $$x - y = -1$$** - From $$x + y = 3$$ and $$x - y = -1$$: $$\begin{cases} x + y = 3, \\ x - y = -1. \end{cases}$$ Adding these two equations: $$2x = 2 \quad \Rightarrow \quad x = 1.$$ Substituting $$x = 1$$ into $$x + y = 3$$: $$1 + y = 3 \quad \Rightarrow \quad y = 2.$$ ### Final Answer: The solutions to the system are: $$(x, y) = (2, 1) \quad \text{or} \quad (1, 2).$$ Do you want more details or have any questions? Here are five related questions for further exploration: 1. How would the solutions change if the first equation was $$5^{x+y} = 625$$? 2. What if the second equation was $$4^{(x-y)^2 - 2} = 1$$? How would that affect the solutions? 3. Can you solve a similar system where the base of the first equation is not a prime number? 4. What if the exponents were fractional instead of integer values? 5. How does the method change if the equations involve logarithms instead of exponents? **Tip:** When solving systems involving exponents, it helps to express both sides of the equation with the same base whenever possible.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Exponential equations
Logarithms
Systems of equations
Formulas
Exponential equations formula
Logarithmic equations formula
Systems of equations solving methods
Theorems
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Suitable Grade Level
Grades 10-12