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Chapter 8: Problem 178

Solve the linear equation using the general strategy. $$-(w-6)=24$$

Short Answer

Expert verified
The solution is \( w = -18 \).

Step by step solution

01

- Simplify the equation

To eliminate the negative sign, distribute the negative sign inside the parentheses: \[-(w - 6) = -w + 6\]. Now we have the equation:\[-w + 6 = 24\].
02

- Isolate the variable term

Move the constant term (6) to the other side by subtracting 6 from both sides: \[-w + 6 - 6 = 24 - 6\]. This simplifies to:\[-w = 18\].
03

- Solve for the variable

To solve for \(w\), multiply both sides of the equation by -1: \[(-w) \times (-1) = 18 \times (-1)\]. This gives: \[w = -18\].

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Key Concepts

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

General Equation Solving Strategy
When solving linear equations, there is a general approach you can follow to make the process more straightforward. First, simplify both sides of the equation. This can involve combining like terms and simplifying expressions. Then, move all terms containing the variable you're solving for to one side of the equation, and all constant terms to the other. Finally, isolate the variable by performing operations that help you solve for it. Let's break it down step-by-step for better understanding:

1. Simplify both sides of the equation, if needed.
2. Move variable terms to one side and constants to the other.
3. Isolate the variable to solve for it.
Distributive Property
The distributive property is a crucial concept when simplifying equations, especially when dealing with parentheses. This property states that a term outside the parentheses can be distributed to each term inside. The formula looks like this: This property helps in breaking down complex expressions. For example, in the problem step 1 is to distribute the negative sign: This simplifies the equation to using the distributive property removes parentheses, making it easier to work with the equation.
Isolating the Variable
Isolating the variable is a key step in solving linear equations. The goal is to get the variable on one side of the equation by itself. This often involves several smaller steps:
  • First, move constants away from the side containing the variable. You usually do this by adding or subtracting the constant from both sides of the equation.
  • Next, if the variable has a coefficient (a number multiplying it), divide or multiply both sides of the equation by that coefficient to solve for the variable.
In our example:

  • Move the constant term to the other side: -w + 6 - 6 = 24 - 6
  • The equation now simplifies to -w = 18
Finally, multiplying both sides by -1 gives w = -18.

By following these steps, you correctly isolate the variable and find the solution to the equation.

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