Problem 62 - Prealgebra and Introductory Algebra

Question

Is \(-4\) a solution of the equation \(6 m=-24 ?\)

Step-by-Step Solution

Answer

Yes, -4 is a solution of the equation \(6 m= -24\).

Step 1: Identify the given equation and the value to check

The equation provided is \(6m = -24\), and it needs to be checked if -4 is a solution of this equation.

Step 2: Substitute the value in the given equation

Replace the variable \(m\) in the equation with -4 to get \(6*(-4) = -24\).

Step 3: Simplify

Solve the left side of the equation which becomes \(-24\). The equation becomes \(-24 = -24\).

Step 4: Verify

Since both sides of the equation are equal, the equation is valid. Therefore, -4 is indeed a solution of the given equation.

Key Concepts

Substitution Method

In algebra, the substitution method is a key technique used to determine if a specific value solves an equation. It involves replacing variables with given numbers and checking if the resulting statement is true. In our exercise, you need to check if \(-4\) satisfies the equation \(6m = -24\).

To begin, identify what you're aiming to do: substitute the variable, in this case, the letter \(m\), with \(-4\). This changes your original equation into \(6(-4) = -24\). It's the crux of the substitution method - a straightforward swap of letters for numbers!

By putting the value in place of the variable, you shift your focus from dealing with variables to working with a concrete number. This makes solving the equation much simpler. All you have to do then is perform basic arithmetic to see if everything checks out. This method is quite intuitive and feels like piecing together a puzzle correctly.

Verification of Solutions

Verification of solutions is an essential step to confirm that the answer you've calculated is correct. After applying a method like substitution, you don't just stop there. You need to be absolutely sure your results are correct. Let’s consider the equation \(6m = -24\) again, where you've substituted \(m\) with \(-4\).

Once you substitute, it simplifies to \(-24 = -24\). The goal here is to ensure both sides of your equation match perfectly.

• If they do, like in our case, then \(-4\) is a correct solution.
• If they don't, then there might be a mistake somewhere, and \(-4\) wouldn't be your solution.

The entire verification approach acts as quality control, ensuring that the steps you've taken and the computations you've completed are accurate. Verification is important as it gives confidence that you’ve understood and applied the method correctly.

Equation Simplification

Equation simplification is a process of reducing an equation to its simplest form, making it easier to understand and solve. It involves performing operations necessary to make the equation as straightforward as possible. In our exercise example, once you substitute \(m = -4\) into \(6m = -24\), the equation becomes \(6(-4) = -24\).

The next step involves simplifying the left side of the equation, which is \(6 \times -4\). This arithmetic operation results in \(-24\). Simplification helps present both sides of the equation in a clean, comparable manner - \(-24 = -24\).

• With simplification, you break down the steps.
• Each simplification step aims to reduce complexity and enhance clarity.
• For instance, by turning \(6(-4)\) directly into \(-24\), you isolate the solution evaluation.

It ensures the whole process is transparent and manageable, bridging your original equation to an easier and more understandable form.