Problem 136 - Prealgebra and Introductory Algebra

Question

Is \(-8\) a solution of the equation \(-12=m-4 ?\)

Step-by-Step Solution

Answer

Yes, -8 is a solution of the equation -12 = m - 4.

Step 1: Substitute

First step, substitute m in the equation with -8, which makes the equation look like this: -12 = -8 - 4.

Step 2: Simplify

Next, simplify the right-hand side of the equation: -8 - 4 = -12.

Step 3: Check Equality

Check if both sides of the equation are equal. Since both sides of the equation equal -12, it means that -8 is a solution for the equation.

Key Concepts

Substitution Method

The substitution method is a fundamental concept in algebra. It involves replacing a variable in an equation with a given value to test if it makes the equation true. In this particular exercise, the substitution method is used to verify if the number \(-8\) can satisfy the equation \(-12 = m - 4\).

• To do this, substitute \(-8\) in place of \(m\) in the equation.
• The equation then becomes \(-12 = -8 - 4\).
• This is the starting point for further simplification and verification.

The substitution method is helpful for checking possible solutions quickly. Use it whenever you're tasked with determining if a specific number is a solution to an equation.

Solution Verification

Once you've substituted the variable with the proposed solution, the next step is to verify it. Solution verification confirms that both sides of the equation are indeed equal after substitution. Here's how it works:

• After substituting, simplify the equation as much as possible.
• Then check if the equation holds true. Both sides of the equation must be equal.
• In this exercise, once you simplify \(-8 - 4\), you get \(-12\).

If the two sides of the equation are the same, the proposed solution is verified and correct. If not, the number is not a solution. This step is crucial for ensuring accuracy in algebraic solutions.

Simplifying Expressions

Simplifying expressions is an integral part of solving equations. This often involves combining like terms or performing arithmetic operations. The right-hand side of the equation \(-12 = -8 - 4\) was simplified by adding the numbers together.

• Start by calculating \-8 - 4\, which simplifies to \-12\.
• Once simplified, compare this result against the left-hand side.

Simplifying expressions helps in making complex equations more manageable and easier to solve. Ensure every step is done carefully to maintain equality and reach the correct conclusion. Understanding how to simplify properly can significantly improve one's ability to solve algebraic problems efficiently.