Problem 14 - Fundamentals of Mathematics

Question

Verify that each given value is a solution to the given equation. $$2 m-1=1, m=1$$

Step-by-Step Solution

Answer

Yes, \( m = 1 \) is a solution.

Step 1: Substitute the given value into the equation

We have the equation \( 2m - 1 = 1 \) and the value \( m = 1 \). Substitute \( m = 1 \) into the equation: \( 2(1) - 1 = 1 \).

Step 2: Simplify the left side of the equation

Perform the multiplication on the left side: \( 2 \times 1 = 2 \). So, the equation becomes \( 2 - 1 = 1 \).

Step 3: Solve by simplifying further

Subtract \( 1 \) from \( 2 \) on the left side: \( 2 - 1 = 1 \), which simplifies to \( 1 = 1 \).

Step 4: Verify the result

Since the left side \( 1 \) and the right side \( 1 \) are equal, \( m = 1 \) is indeed a solution to the equation \( 2m - 1 = 1 \).

Key Concepts

Equation Verification

Equation verification is a crucial part of solving mathematical equations. It involves checking whether a given value satisfies an equation. This is essential to ensure the correctness of the solution. Verification is the final step where you confirm that the work done is correct.

For example, in our exercise, we verify whether the value \( m = 1 \) is a solution for the equation \( 2m - 1 = 1 \).

• First, substitute the given value into the equation.
• Then, simplify both sides of the equation.
• Finally, check if both sides are equal. If they are, the solution is verified.

Verification helps in ensuring that no errors occur during the solution process. It can require repetition of previous steps to check the correctness of computations.

Substitution Method

The substitution method is one of the main techniques used to solve equations. It involves replacing the variable in an equation with a given value or expression. It's a simple but powerful tool, especially when you need to verify solutions.

In our exercise, to apply the substitution method, use the following steps:

• Identify the variable in the equation. Here, it is \( m \).
• Replace the variable with its given value, which is \( m = 1 \) in our example.
• Substitute directly into the equation: \( 2(1) - 1 = 1 \).

This transforms the original equation into a numerical equation. Substitution simplifies both understanding and solving equations, making it an invaluable approach. It allows one to directly test whether a proposed solution is correct.

Simplification Steps

Simplification is the process of reducing an equation to its simplest form. This involves performing basic arithmetic operations and ensuring both sides of the equation are as simple as possible. Simplification helps make equations easier to solve and verify.

To simplify the equation \( 2(1) - 1 = 1 \) during the verification process:

• Perform the multiplication on the left side: \( 2 \times 1 = 2 \).
• Subtract the number \( 1 \) from the result: \( 2 - 1 = 1 \).

The equation now reads \( 1 = 1 \), which is the simplest form and shows a balanced equation.

By following these steps, you smoothly reduce complex expressions. It is a fundamental skill in algebra that aids in solving and checking equations accurately.