Problem 56 - A Graphical Approach to College Algebra

Question

Solve each equation. $$\left(\frac{3}{5}\right)^{-x}=\left(\frac{9}{25}\right)^{1-5 x}$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{2}{9}\).

Step 1: Understand the Problem

The given equation is \(\left(\frac{3}{5}\right)^{-x}=\left(\frac{9}{25}\right)^{1-5x}\). Our goal is to solve for \(x\). This requires making the bases on both sides of the equation the same and then solving for the variable.

Step 2: Express Each Side with a Common Base

Notice that \(\frac{9}{25} = \left(\frac{3}{5}\right)^2\). Use this fact to rewrite the right side of the equation: \[ \left(\frac{3}{5}\right)^{-x} = \left(\left(\frac{3}{5}\right)^2\right)^{1-5x} \].

Step 3: Simplify the Right Side Using Exponent Laws

Apply the exponentiation law \((a^m)^n = a^{m \cdot n}\) to the right side: \[ \left(\frac{3}{5}\right)^{-x} = \left(\frac{3}{5}\right)^{2(1-5x)} \]. Calculate the exponent: \(2(1-5x) = 2 - 10x\), so the equation becomes \(\left(\frac{3}{5}\right)^{-x} = \left(\frac{3}{5}\right)^{2-10x}\).

Step 4: Set Exponents Equal to Each Other

Since the bases are equal, set the exponents equal: \(-x = 2 - 10x\).

Step 5: Solve the Linear Equation

Rearrange the equation to find \(x\). Add \(10x\) to both sides to get \(9x = 2\). Then divide both sides by 9: \[ x = \frac{2}{9} \].

Key Concepts

Solving Equations

When tackling equations, your goal is to find the value of the unknown variable that makes the equation true. Here’s a simple way to think about it:

• Look at each side of the equation. They need to balance each other, just like the two sides of a scale.
• Identify what operations are performed on the variable.
• Reverse these operations to isolate the variable (doing the same thing to both sides, like adding, subtracting, multiplying, or dividing).

In our example, the equation is \(\left(\frac{3}{5}\right)^{-x}=\left(\frac{9}{25}\right)^{1-5 x}\). Both sides need to equal each other, so our main act is to manipulate both sides until they reveal the same base expression. Once the bases are identical, you only need to consider the exponents.

Exponent Laws

Exponent laws help simplify expressions involving powers and make it easier to solve equations like the one given in our exercise. A few essential exponent rules are:

• Product of Powers Rule: \(a^m \cdot a^n = a^{m+n}\)This rule allows you to combine powers of the same base by adding their exponents.
• Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)Use this rule to multiply exponents when a power is raised to another power.
• Negative Exponent Rule: \(a^{-m} = \frac{1}{a^m}\)This rule demonstrates how to convert negative exponents to positive by taking their reciprocal.

In our exercise, the Power of a Power Rule is key. Transforming \(\left(\frac{3}{5}\right)^{2(1-5x)}\) helps in simplifying the equation for solving.

Linear Equations

Linear equations are the simplest type of algebraic equations. They form a straight line when graphed and have the general form \(ax + b = c\). To solve these equations, follow these steps:

• Rearranging: Bring all terms involving the variable to one side and constants to the other side of the equation.
• Isolation: Isolate the variable by reversing operations (additive inverse or multiplicative inverse).
• Solution: Simplify to find the value of the variable.

In the final step of our original exercise, we had a linear equation \(-x = 2 - 10x\). Solving it requires reorganizing to \(9x = 2\), followed by dividing both sides by 9 to get\(x = \frac{2}{9}\). This clear and straightforward manipulation reaches the solution efficiently.