Problem 76 - Elementary and Intermediate Algebra

Question

Check to determine whether 4 is a solution of $$ 3(m-8)+2 m=4-(m+2) $$

Step-by-Step Solution

Answer

No, 4 is not a solution as both sides of the equation are not equal when substituting 4 for \( m \).

Step 1: Substitute m with 4

To verify if 4 is a solution, substitute 4 for \( m \) in the equation. This gives us: \[ 3(4-8) + 2(4) = 4 - (4+2) \]

Step 2: Simplify Left Side

Simplify the expression on the left side of the equation: - Begin with the parentheses: \( 4 - 8 = -4 \), so the expression becomes \[ 3(-4) + 2(4) \]- Multiply the terms: \( 3(-4) = -12 \) and \( 2(4) = 8 \)- Combine the results: \( -12 + 8 = -4 \)

Step 3: Simplify Right Side

Simplify the expression on the right side: - Start with the parentheses: \( 4 + 2 = 6 \), leading to \[ 4 - 6 = -2 \].

Step 4: Compare Both Sides

Compare the simplified versions of both sides of the equation:- Left side is \( -4 \)- Right side is \( -2 \)Since both sides are not equal, \( 4 \) is not a solution.

Key Concepts

Substitution Method

The Substitution Method is a powerful technique in algebra used to determine if a specific value satisfies an equation. It involves replacing the variable in the equation with a given number. This allows us to see if setting the variable equal to this number results in a true statement. In our exercise, we want to check if 4 is a solution to the equation:

• Begin by substituting 4 for the variable \( m \) in the equation. This step is crucial as it transforms the algebraic equation into a numerical one, making it easier to evaluate.
• Once substituted, the equation \[ 3(m-8)+2m = 4-(m+2) \] becomes \[ 3(4-8) + 2(4) = 4 - (4+2) \].

By doing this, we have taken the first step to verify if 4 is a viable solution for the original equation.

Simplifying Expressions

Simplifying Expressions is a key step in algebra that makes equations easier to understand and solve. Once you've substituted in a value, the next step involves breaking down the equation into simpler parts:

• The left side of the equation, after substitution, is \[ 3(4-8) + 2(4) \].
• First, solve within the parentheses, which simplifies to \( 4 - 8 = -4 \).
• This leads to \[ 3(-4) + 2(4) \].
• Multiply the terms: \( 3 \times -4 = -12 \) and \( 2 \times 4 = 8 \), providing us with \[ -12 + 8 \].
• Adding these results, we get \( -4 \).

On the right side:

• We simplify by tackling the parentheses: \( 4 + 2 = 6 \), leaving \[ 4 - 6 \],
• which simplifies to \( -2 \).

By simplifying both sides, we are able to directly compare the results.

Equation Verification

Equation Verification is the final step in confirming if a substituted value is indeed a solution to the equation. After simplifying both sides of our equation, the comparison is straightforward:

• The left side simplified to \( -4 \).
• The right side simplified to \( -2 \).

To verify if a given number is a solution, both sides of the equation must be equal. In this case, since \( -4 \) is not equal to \( -2 \), the number 4 is not a solution to the original equation. This method ensures that the solution satisfies the equation completely, providing confidence in the correctness of the solution.