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Chapter 6: Problem 76

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

Short Answer

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

Step by step solution

01

Substitute m with 4

To verify if 4 is a solution, substitute 4 for \( m \) in the equation. This gives us: \[ 3(4-8) + 2(4) = 4 - (4+2) \]
02

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: \( -12 + 8 = -4 \)
03

Simplify Right Side

Simplify the expression on the right side: - Start with the parentheses: \( 4 + 2 = 6 \), leading to \[ 4 - 6 = -2 \].
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Evaluate each expression. $$ 7^{0} $$

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