Problem 37 - Algebra 2 and Trigonometry

Question

Solve each equation and check. \(e^{2 x+2}=e^{x-1}\)

Step-by-Step Solution

Answer

The solution is \(x = -3\).

Step 1: Set the Exponents Equal

Since the bases of the exponential expressions are the same, we can set their exponents equal to each other. So, we have:\[2x + 2 = x - 1\]

Step 2: Solve for x

To find the value of \(x\), subtract \(x\) from both sides:\[2x + 2 - x = x - 1 - x\]This simplifies to:\[x + 2 = -1\]Subtract 2 from both sides to isolate \(x\):\[x = -1 - 2\]Therefore, \(x = -3\).

Step 3: Check the Solution

Substitute \(x = -3\) back into the original equation to verify the solution:Substitute into the left-hand side:\[e^{2(-3) + 2} = e^{-6 + 2} = e^{-4}\]Substitute into the right-hand side:\[e^{-3 - 1} = e^{-4}\]Both sides are equal to \(e^{-4}\), confirming the solution is correct.

Key Concepts

Exponents

When we talk about exponents, we're discussing a mathematical way of representing repeated multiplication. An exponent tells us how many times to multiply a base number by itself. For example, in the expression \(e^3\), the base \(e\) is multiplied by itself three times: \(e \times e \times e\). Exponents are important because they simplify expressions and equations, especially when dealing with large numbers. They are a fundamental part of many mathematical disciplines, including calculus and algebra. When dealing with exponential equations, like \(e^{2x+2} = e^{x-1}\), we're not only comparing sizes but also manipulating these expressions in a structured way by operating on their exponents.

Equation Solving

Solving an equation means finding the value of the variable that makes the equation true. For an exponential equation, this typically involves manipulating the equation so that you can compare and equate the exponents directly.
In our case, with \(2x + 2 = x - 1\), the goal was to find the value of \(x\). We did this by isolating \(x\) through a series of algebraic steps:

• Subtract \(x\) from both sides, reducing the equation to \(x + 2 = -1\).
• Then, by subtracting 2 from both sides, we further isolated \(x\), getting \(x = -3\).

These steps are fundamental in mathematics because they demonstrate how to manipulate and transform equations into simpler forms to reveal the unknowns we seek.

Base Equality

An important concept in solving exponential equations is base equality. This means that if two exponential expressions have the same base, their exponents must be equal for the overall expressions to be equal.
In the equation \(e^{2x+2} = e^{x-1}\), both sides have the same base \(e\), so we equated the exponents, setting \(2x + 2 = x - 1\). By focusing on the equality of the exponents, we reduced the problem to an algebraic one, allowing us to solve for \(x\). This principle simplifies the solving process considerably and is a powerful tool in dealing with exponential equations across various mathematical applications.

Verification of Solutions

Once a solution is found, it is crucial to verify that it satisfies the original equation. Verification helps ensure the solution is correct and confirms that no mistakes were made during calculation. For our equation, after finding \(x = -3\), substituting back into the original:

• Left side: \(e^{2(-3)+2} = e^{-4}\)
• Right side: \(e^{-3-1} = e^{-4}\)

Both sides equate to \(e^{-4}\), confirming our solution is correct. Verification not only proves the accuracy of our work but fosters a deeper understanding and confidence in the solving process. Ensuring both sides of the initial equation are equivalent after substituting the found solution is a critical step in problem-solving.