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Chapter 2: Problem 127

Solve each equation. Be sure to check each answer. $$-27 w=81$$

Short Answer

Expert verified
The solution to the equation \(-27 w = 81\) is \(w = -3\).

Step by step solution

01

Isolate the Variable

To isolate 'w' on one side, divide both sides of the equation by -27. This operation yields: \(w = \frac{81}{-27}\).
02

Simplify

Simplify the right side of the equation to get: \(w = -3\).
03

Check the Answer

Substitute 'w' in the original equation with -3: \(-27*(-3) = 81\), which turns out to be true. Therefore, the solution is correct.

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Key Concepts

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

Solving Equations
Solving equations is a fundamental aspect of algebra and involves finding the value of a variable that makes the equation true. In our example with the equation \(-27w = 81\), the goal is to find the value of \(w\) such that the left side of the equation equals the right side. Solving equations generally involves a series of steps aimed at uncovering the unknown variable.

Here are some key points to consider when solving equations:
  • Understand the equation's structure: Identify what operation is being applied to the variable.
  • Perform the inverse operation: In our example, since the variable is being multiplied by \(-27\), we perform the inverse operation, division, to both sides of the equation to eliminate \(-27\).
  • Verify your solution: Always substitute your solution back into the original equation to ensure it holds true.
When you correctly perform these steps, you will find that \(w = -3\), and a quick check confirms the solution is correct.
Variable Isolation
Variable isolation is the step in equation solving that aims to get the variable by itself on one side of the equation. This technique makes it easier to see the value of the variable. In the equation \(-27w = 81\), isolating \(w\) requires us to remove \(-27\) from \(w\) by performing the opposite operation.

For effective variable isolation:
  • Identify operations affecting the variable: Here, \(w\) is being multiplied by \(-27\).
  • Use inverse operations to remove them: Divide both sides by \(-27\), which gives us \(w = \frac{81}{-27}\).
  • Perform these operations carefully to maintain the equation's balance.
Once the variable is isolated (\(w\) is left alone), it becomes straightforward to identify \(w = -3\). Ensuring the variable is isolated correctly is crucial for solving any algebraic equation.
Simplifying Equations
Simplifying equations is about making an equation easier to read and solve by reducing it to its simplest form. After isolating the variable in an equation, simplifying helps you get the final, clear-cut value of the variable.

Let's simplify step-by-step following our example:
  • Once you isolate \(w\), you are left with \(w = \frac{81}{-27}\).
  • The task is to simplify \(\frac{81}{-27}\). Negative signs in a fraction can be perplexing, but you only have to remember that dividing by a negative number inverts the sign.
  • Divide 81 by 27, which gives 3, then apply the negative sign: hence, \(w = -3\).
Simplification not only ensures accuracy but also allows for a clearer understanding of the results. Consistency in simplifying equations across different problems helps in developing a structured approach to solving mathematical problems.

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