Problem 24 - Algebra 2 and Trigonometry

Question

Solve each equation and check. \(49^{x}=7^{3 x+1}\)

Step-by-Step Solution

Answer

The solution is \(x = -1\), and substituting it back verifies the equation.

Step 1: Express Both Sides with Base 7

Convert the left side of the equation to a power of 7. We know that \(49 = 7^2\), so we rewrite the equation as \((7^2)^{x} = 7^{3x+1}\).

Step 2: Apply Exponent Rule

Using the exponent multiplication rule \((a^m)^n = a^{mn}\), the left side becomes \(7^{2x}\). So the equation is now \(7^{2x} = 7^{3x+1}\).

Step 3: Set the Exponents Equal

Since the bases are equal, set the exponents equal: \(2x = 3x + 1\).

Step 4: Solve for x

Rearrange the equation to solve for \(x\) by subtracting \(3x\) from both sides to get \(-x = 1\). Then multiply both sides by \(-1\) to find \(x = -1\).

Step 5: Check the Solution

Substitute \(x = -1\) back into the original equation to verify the solution. The left side becomes \(49^{-1} = \frac{1}{49}\), and the right side becomes \(7^{3(-1)+1} = 7^{-2} = \frac{1}{49}\). Both sides are equal, confirming \(x = -1\) is correct.

Key Concepts

Understanding Base Conversion in Exponential Equations

Base conversion is an essential step when tackling exponential equations. It simplifies expressions by ensuring that both sides of the equation have the same base. For instance, in the equation \(49^x = 7^{3x+1}\), the base on the left side is 49, while on the right, it is 7. To deal with this, we utilize a clever trick: convert \(49\) into a power of \(7\). We know that \(49\) can be expressed as \(7^2\). Thus, \(49^x = (7^2)^x\).

By converting \(49\) to \(7^2\), the equation becomes \((7^2)^x = 7^{3x+1}\), and now both sides have the same base of 7. This conversion simplifies the equation significantly and sets the stage for using exponent rules effectively.

Applying Exponent Rules for Simplification

Exponent rules help simplify expressions and solve exponential equations.

• One crucial rule is the power of a power: \((a^m)^n = a^{mn}\).
• This rule tells us how to deal with exponents raised to another exponent by multiplying them together.

In the equation \((7^2)^x = 7^{3x+1}\), we apply this rule to the left side, treating \(7^2\) raised to the power of \(x\), simplifying to \(7^{2x}\). Now the equation reads \(7^{2x} = 7^{3x+1}\), and since both have the same base, solving becomes easier. These rules are vital tools in algebraic manipulation, making complex problems much more manageable.

Algebraic Manipulation to Find the Solution

With bases matched and exponent rules applied, we reach a critical step: algebraic manipulation. Algebraic manipulation involves rearranging equations to isolate and solve for variables. In our example, once simplified, the equation is \(7^{2x} = 7^{3x+1}\). Since both sides now have base 7, we can directly equate their exponents: \(2x = 3x + 1\).

To find \(x\), manipulate as follows:

• Subtract \(3x\) from both sides: \(2x - 3x = 1\).
• This simplifies to \(-x = 1\).
• Multiply both sides by \(-1\) to solve for \(x\): \(x = -1\).

This process highlights how manipulation through addition, subtraction, and multiplication can uncover variable values in exponential equations.

Solution Verification to Confirm the Correctness

After determining a solution for an equation, verifying its correctness is crucial. Verification ensures that no errors were made during calculation. Once we calculated \(x = -1\) in our problem, the next step is to check the original equation. Substitute \(x = -1\) back into the original equation:

• The left side becomes \(49^{-1} = \frac{1}{49}\).
• The right side becomes \(7^{3(-1)+1} = 7^{-2} = \frac{1}{49}\).

Since both sides equate to \(\frac{1}{49}\), it confirms the solution is correct. This verification step validates our computations and provides confidence in the final answer, cementing \(x = -1\) as accurate.