Problem 29 - Algebra 2 and Trigonometry

Question

If \(3^{a+1}=x\) and \(3^{a}=y,\) express \(y\) in terms of \(x\)

Step-by-Step Solution

Answer

The expression for \(y\) in terms of \(x\) is \(y = \frac{x}{3}\).

Step 1: Understand the Given Equations

We have two equations: \(3^{a+1}=x\) and \(3^a=y\). We need to express \(y\) in terms of \(x\).

Step 2: Simplify the Equation with Exponents

Rewrite the equation \(3^{a+1}=x\) as \(3^a \cdot 3^1 = x\) which is \(3^a \cdot 3 = x\).

Step 3: Isolate \(3^a\) from the Simplified Equation

Rearrange the equation from Step 2 to solve for \(3^a\): \(3^a = \frac{x}{3}\).

Step 4: Equate \(y\) to \(3^a\)

Since \(3^a = y\), substitute \(3^a = \frac{x}{3}\) from Step 3 into this equation. Thus, \(y = \frac{x}{3}\).

Key Concepts

Exponential Expressions

Exponential expressions involve a base that is raised to a power or an exponent. In our exercise, the base is 3 and the exponents vary between expressions to define different quantities. Understanding how exponents work is crucial, as they determine the growth of the base number through multiplication.

For our purposes:

• In the expression \(3^{a+1}\), the exponent \(a+1\) indicates that the base 3 is multiplied by itself \(a+1\) times.
• Similarly, \(3^a\) implies the base 3 is used as a factor \(a\) times.

These expressions show how we can manipulate the power of a base to express equations—and even variables—differently. For instance, adding exponents \(a+1\) or \(a\) impacts the resulting value significantly. When transitioning between these expressions, it's important to remember key exponential rules, such as that \(3^{a+1} = 3^a imes 3\). This understanding is pivotal when you simplify or change forms of exponential expressions in algebraic contexts.

Problem Solving in Algebra

Problem-solving in algebra is all about understanding the relationships between different expressions and manipulating them. In our scenario, two equations are given with exponential terms: \(3^{a+1} = x\) and \(3^a = y\). The real task lies in expressing one term (\(y\)) in terms of another (\(x\)).

This involves a simplified step-by-step approach:

• First, break down complex expressions. Rewrite \(3^{a+1}\) as \(3^a \times 3\). This hot new form helps illuminate internal relationships.
• Compare and isolate components to form new equations. You isolate \(3^a\) by dividing \(x\) by 3, following specific algebraic tactics to readily express \(y\) in the form of \(\frac{x}{3}\).

By tackling problems methodically and checking your work frequently, you develop a knack for solving algebraic expressions efficiently. Problem-solving in algebra is about finding clarity in complex relationships, making this journey through exponents educational and practical.

Equation Transformation

Equation transformation is the art of reshaping equations to make them more useful or solve them. It often involves using known algebraic rules to progress from a complex state to a more manageable solution.

In this exercise, the equation \(3^{a+1} = x\) is transformed by recognizing that \(3^{a+1}\) is the same as \(3^a \cdot 3\). Through this transformation, a strategic insight allows solving for \(3^a\):

• Transform \(3^{a+1}\) into \(3^a \times 3\) to link it to \(y\).
• Isolate \(3^a\) by dividing both sides by 3:\[3^a = \frac{x}{3}\].

Once transformed, it's easy to express \(y\) as \(\frac{x}{3}\) because \(3^a = y\). Equation transformation is essential because it enables direct comparison and understanding, altering perspectives to highlight obvious solutions unbeknownst in the original form. Understanding these transformations saves time and boosts comprehension in algebraic operations.