Problem 111 - Prealgebra and Introductory Algebra

Question

Is 8 a solution of the equation \(\frac{m}{-4}=\frac{-16}{m} ?\)

Step-by-Step Solution

Answer

Yes, 8 is a solution to the given equation.

Step 1: Substitute the value of m

Replace m by 8 in the given equation. After substitution, the equation will be: \(\frac{8}{-4}=\frac{-16}{8}\).

Step 2: Simplify

Simplify both sides of the equation. In the left side, \(8/-4=-2\) and on the right side \(-16/8 = -2 \). So, the simplified equation will be -2=-2.

Step 3: Verify

Since after substitution and simplification, both sides of the equation equal each other, therefore we can say that 8 is indeed a solution to the given equation.

Key Concepts

Equation Solving

The process of solving an equation involves finding the value of the variable that makes the equation true.
In simple terms, we want to find the number that can replace the variable in the equation to make both sides equal.
In the given exercise, the equation is \( \frac{m}{-4}=\frac{-16}{m} \).
To check if 8 is a solution, we need to determine if substituting 8 for \( m \) satisfies the equation.

Substitution Method

The substitution method is a straightforward and commonly used technique in equation solving.
It involves replacing the variable in an equation with a given number to see if the equation holds true.
For the problem at hand, we're checking whether 8 is a solution.

• Substitute 8 in place of \( m \) in the equation: \( \frac{8}{-4} = \frac{-16}{8} \).
• By doing this, we transform the equation into a numerical form, allowing us to work with plain numbers.

By substituting, we essentially test if our guess (in this case, 8) is correct.

Simplification

Once substitution is complete, the next step is simplification.
Simplification means reducing the equation to its simplest form by performing any arithmetic operations required.
In our example:

• On the left side: \( \frac{8}{-4} = -2 \).
• On the right side: \( \frac{-16}{8} = -2 \).

Both sides simplify to the same value, \(-2\), which is an important indicator during equation solving.
When both sides match after simplification, it suggests our substituted value might be correct.

Verification Process

Verification is the final and crucial step in solving equations.
It involves confirming that the simplified equation holds true for the value we substituted.
In this case, after substitution and simplification, both sides of the equation equate to \(-2\).
Since they are equal, we verify that our substitution was correct.
This confirms that 8 is indeed a solution to the equation \( \frac{m}{-4}=\frac{-16}{m} \).

• This step ensures that no arithmetic or substitution errors have occurred.
• It reassures that our solution is valid and we successfully solved the equation.

Verification acts as a completeness check for your solution.