Problem 123 - Thomas Calculus

Question

\begin{equation}y^{x}=x^{3} y\end{equation}

Step-by-Step Solution

Answer

One solution is \( x = 1 \) and \( y = 1 \).

Step 1: Analyze the Equation

The given equation is \( y^x = x^3 y \). Our goal is to solve for one of the variables, typically by expressing one variable in terms of the other.

Step 2: Rearrange the Equation

Rearrange the equation to see if simplification is possible. Start by dividing both sides of the equation by \( y \), assuming \( y eq 0 \): \[ \frac{y^x}{y} = x^3 \].

Step 3: Simplify the Expression

Simplify the left side of the equation: \[ y^{x-1} = x^3 \]. This implies that \( y^{x-1} \) is equal to \( x^3 \).

Step 4: Consider Specific Solutions

Let's consider what values of \( x \) and \( y \) satisfy this new equation. For simplicity, test if there are integer solutions such as \( y = 1 \) or \( x = 1 \).

Step 5: Check Integer Solution: y = 1

Try \( y = 1 \), then the equation becomes:\[ 1^{x-1} = x^3 \] which simplifies to \( 1 = x^3 \). The only real number \( x \) that satisfies this is \( x = 1 \).

Step 6: Verify Solution

Verify the solution \( x = 1, y = 1 \) in the original equation:\( 1^1 = 1^3 \times 1 \) which results in \( 1 = 1 \). Thus, the solution satisfies the initial equation.

Key Concepts

Exponential Equations

Exponential equations are equations in which a variable appears in the exponent. The standard form is like our initial equation: \( y^x = x^3 y \). This format indicates that the equation balances powers of variables on both sides.
Understanding exponential behavior:

• Growth or decay involving exponential terms is typically rapid, making such equations interesting yet complex.
• Exponential functions often model real-world phenomena like population growth and radioactive decay.

When dealing with exponential equations, finding integer solutions becomes an interesting challenge. In our example, this was tackled by rearranging and simplifying the equation.

Integer Solutions

Integer solutions refer to solutions of equations where the variables take on whole number values (integers). For equations like \( y^{x-1} = x^3 \), finding such solutions involves testing feasible whole number values.

• An integer solution is often preferred because it simplifies interpretation and verification.
• Checking integer solutions can be done efficiently with small numbers by logical substitution.

In the given solution process, integers were tested directly to find the pair \( x = 1 \), \( y = 1 \) as a feasible solution.
Detecting such solutions can sometimes suggest symmetries or special cases that are not immediately obvious from the algebraic manipulation alone.

Simplification of Expressions

Simplifying expressions is a key step in solving equations. It helps to highlight relationships between variables and objectives.

• In our problem, the simplification was achieved by dividing both sides by \( y \), resulting in a clearer equation: \( y^{x-1} = x^3 \).
• This change reduces complexity and pares the equation down to its essential components.

Such operations aim to convert an equation into a more tangible form. This process oftentimes involves identifying common terms, like reducing powers, which immensely helps in understanding the core relationship of the variables.

Equation Solving Steps

Solving an equation systematically involves multiple steps with troubleshooting and verification:

• **Rearrangement:** Start by ensuring the equation is in a solvable form.
• **Simplification:** Simplify the expressions, as seen with our division step which gave \( y^{x-1} = x^3 \).
• **Testing values:** Check for specific solutions, often starting with simple or integer values.
• **Verification:** Always substitute back to ensure solutions meet the original equation's needs.

For example, the solution tackled possible integer values of \( x \) and \( y \). After simplification and testing, \( x = 1 \) and \( y = 1 \) were found to satisfy the equation. It's vital to validate solutions by reapplying them in the original equation to confirm accuracy.