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Magazine Chapter 9: Problem 41
Solve the given nonlinear system. $$ \left\\{\begin{array}{l} x=3^{y} \\ x=9^{y}-20 \end{array}\right. $$
Short Answer
Expert verified
The solution is \((x, y) = (5, \log_3(5))\).
Step by step solution
01
Understand the Equations
The given system consists of two equations: \( x = 3^y \) and \( x = 9^y - 20 \). Our goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously.
02
Express with the Same Base
Recognize that \(9^y\) can be rewritten using powers of 3: \(9^y = (3^2)^y = (3^y)^2\). Substituting \(3^y = x\), we have \(9^y = x^2\).
03
Substitute and Simplify
Substitute \(x = 3^y\) from the first equation into the transformed second equation: \(x = x^2 - 20\). This simplifies to \(x^2 - x - 20 = 0\).
04
Solve the Quadratic Equation
Solve the quadratic equation \(x^2 - x - 20 = 0\) using factoring, the quadratic formula, or any suitable method. Factoring gives \((x - 5)(x + 4) = 0\), so \(x = 5\) or \(x = -4\).
05
Determine Corresponding y Values
Use \(x = 3^y\) to find corresponding \(y\) values for \(x = 5\). There is no real \(y\) for \(x = -4\) since \(3^y\) cannot be negative. Solving \(3^y = 5\) gives \(y = \log_3(5)\).
06
Verify Solutions
Check if these solutions satisfy the second original equation. For \(x = 5\) and \(y = \log_3(5)\), calculate \(9^y = (3^y)^2 = 5^2 = 25\), which gives \(9^y - 20 = 25 - 20 = 5 = x\). Thus, the solution works.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Quadratic Equations
Quadratic equations are vital in understanding nonlinear systems. They involve a variable raised to the second power, often written as \[ ax^2 + bx + c = 0 \]. In the context of the given exercise, the equation \( x^2 - x - 20 = 0 \) is quadratic. Quadratics can have one, two, or no real solutions, illustrated by their graph as a parabola. Solving them can be achieved by:
- Factoring: Breaking down the equation into product of linear factors; e.g., \((x - 5)(x + 4) = 0\).
- Using the Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), applicable when factoring is difficult.
- Completing the Square: Transforming the equation to a perfect square trinomial.
Unraveling Exponential Equations
Exponential equations are equations where variables appear as exponents, like \(x = 3^y\) from our exercise. They feature constant bases raised to variable powers. The key challenge is determining the exponent, which cannot be done with basic algebra alone. Key properties include:
- Comparing Bases: Simplifying and matching the bases to form solvable relationships. For instance, transforming \(9^y\) into \((3^y)^2\).
- Base 3 in Context: The base 3 exponentials linked crucially to our quadratic form, driving solutions by substitution.
Exploring Logarithmic Functions
Logarithmic functions are inverses of exponential functions, crucial for solving equations like \(3^y = 5\). They express the power needed to get a number using a base, here \(\log_3(5)\). Important properties include:
- Logarithms as Inverse Operations: \(y = \log_b(A)\) means \(b^y = A\), useful for finding exponents in exponential equations.
- Logarithmic Properties: Including product, quotient, and power rules, streamlining complex calculations.
