Problem 107 - Introductory Algebra for College Students

Question

Determine whether the given number is a solution of the equation. $$\frac{5 m-1}{6}=\frac{3 m-2}{4} ;-4$$

Step-by-Step Solution

Answer

-4 is indeed a solution of the given equation.

Step 1: Substitute the Value

Start by substituting the value -4 for m into the equation, which yields \(\frac{5(-4) -1}{6} = \(\frac{3(-4) -2}{4}\).

Step 2: Simplify Each Side of the Equation

Next, simplify each expression on both sides of the equation. By doing so, you get \(\frac{-21}{6} = \(\frac{-14}{4}\). Further simplification gives \(-3.5 = -3.5\).

Step 3: Verifying the Solution

Lastly, it's necessary to verify whether both sides of the equation are equal, if they are, this means that -4 is a solution of the given equation. In this case, since -3.5 = -3.5, -4 is indeed a solution.

Key Concepts

Substitution Method

Understanding the substitution method is crucial in algebra, particularly when you're tasked with determining whether a specific value is a solution to an equation. This method involves replacing a variable with a given number to evaluate the outcome.

For example, when you're given the equation \( \frac{5m - 1}{6} = \frac{3m - 2}{4} \) and asked if -4 is a solution, the substitution method allows you to 'plug in' this value in place of the variable m. Here, you would substitute -4 for every instance of m in the equation, resulting in \( \frac{5(-4) - 1}{6} = \frac{3(-4) - 2}{4} \).

Using substitution systematically for checking solutions teaches you to tackle equations step by step and avoids confusion, enabling you to focus on simplifying the expression next.

Simplifying Expressions

After applying the substitution method to an algebraic equation, the next step is to simplify the expressions. Simplifying is the process of rewriting an expression in a more manageable or compact form without changing its value. It typically involves carrying out arithmetic operations and reducing fractions where possible.

In our example, once we substitute -4 into the equation, we then have to perform the arithmetic within the numerators to get \( \frac{-20 - 1}{6} = \frac{-12 - 2}{4} \), which further simplifies to \( \frac{-21}{6} = \frac{-14}{4} \). From here, we divide to obtain the simplest form of these fractions, which in this case both simplify to -3.5.

Helpful Tips for Simplification:

• Always perform operations within parentheses first, following the order of operations.
• Combine like terms where possible to reduce the complexity of the expression.
• Reduce fractions to their simplest form to easily compare expressions.

Learning to simplify correctly is an indispensable skill as it can turn complex, daunting problems into more understandable and solvable ones.

Verifying Solutions

Verifying solutions in algebra serves as a proof to confirm that your answer is correct. After substituting values and simplifying the expressions, compare the results from both sides of the equation. If the values are equal, your solution is verified correct.

In our working scenario, after simplification, we arrived at -3.5 for both sides of the equation. Verifying that these two values are indeed equal, confirms that -4 is a solution for the given equation.

Verification is a fundamental step because it ensures the accuracy of your work. It not only confirms the validity of your solution but also enhances your understanding of the problem-solving process. Additionally, regular practice in verifying solutions will improve your attention to detail and help prevent common mistakes in algebra.

Steps to Ensure Accurate Verification:

• Carefully compare the simplified results from both sides of the equation.
• Recheck your substitution and simplification steps for any possible errors.
• Confirm that all operations are executed correctly according to the order of operations.

Mastery of verification not only improves your confidence in problem solving but is also essential for tackling more complex algebraic tasks.