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Chapter 2: Problem 50

Solve each equation. $$ -(w+0.2)=0.3(4-w) $$

Short Answer

Expert verified
The solution is \(w = -2\).

Step by step solution

01

Distribute on the Right Side

Begin by distributing the multiplication on the right side of the equation. The equation is given as \(-(w + 0.2) = 0.3(4 - w) \). Distribute the \(0.3\) to both terms inside the parentheses, which results in \(0.3 \cdot 4 - 0.3 \cdot w = 1.2 - 0.3w\). Now the equation looks like this: \(-(w + 0.2) = 1.2 - 0.3w\).
02

Distribute the Negative Sign on the Left Side

Distribute the negative sign on the left side of the equation. The left side is \(-(w+0.2)\), which can be expanded to \(-w - 0.2\). Now the equation is \(-w - 0.2 = 1.2 - 0.3w\).
03

Move Variables to One Side

To isolate the variable \(w\), add \(0.3w\) to both sides of the equation to combine like terms. This gives you \(-w + 0.3w - 0.2 = 1.2\), simplifying to \(-0.7w - 0.2 = 1.2\).
04

Isolate the Variable Term

Add 0.2 to both sides to isolate the term with \(w\): \(-0.7w - 0.2 + 0.2 = 1.2 + 0.2\). This simplifies to \(-0.7w = 1.4\).
05

Solve for the Variable

To solve for \(w\), divide both sides by \(-0.7\): \(w = \frac{1.4}{-0.7}\). Simplifying this division gives you \(w = -2\).

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Key Concepts

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

Distributive Property
When dealing with linear equations, the distributive property is a helpful tool to simplify expressions. You use this property to remove parentheses by distributing a multiplication across terms inside the parentheses. In the original exercise, the distributive property is applied twice: first on the right side
  • Distribute the \(0.3\) over \(4 - w\), leading to \(0.3 \times 4 - 0.3 \times w = 1.2 - 0.3w\).
On the left side:
  • Distribute the negative sign across \(w + 0.2\) to get \(-w - 0.2\).
This step ensures that all distributed terms are multiplied as needed, allowing the equation to be written without parentheses. It lays the foundation for further simplification by making all terms visible and manageable.
Combining Like Terms
After applying the distributive property, the next crucial step is combining like terms. Like terms have the same variables raised to the same power. To simplify the equation, it's vital to bring these terms together. In the context of the problem:
  • The equation transforms into \(-w - 0.2 = 1.2 - 0.3w\).
  • Add \(0.3w\) to both sides to handle the terms involving \(w\), resulting in \(-w + 0.3w - 0.2 = 1.2\), or simplified further to \(-0.7w - 0.2 = 1.2\).
The aim here is to group all variable and constant terms, making it easier to solve the equation. Combining like terms reduces clutter and paves the way for isolating the variable.
Isolating Variables
Isolating the variable is essential for finding its value. This involves getting the variable term by itself on one side of the equation. In the exercise, our goal is to isolate \(w\):
  • Once simplified to \(-0.7w - 0.2 = 1.2\), add \(0.2\) to both sides to eliminate the constant on the variable side, resulting in \(-0.7w = 1.4\).
Isolating the variable sets the stage for solving the equation, allowing a straightforward path to determine its exact value. The trick is carefully and systematically removing any terms interfering with the variable.
Solving for a Variable
Solving for a variable is the final step, where you find its exact value. Once the variable is isolated, the solution becomes apparent. For the current problem with the equation \(-0.7w = 1.4\):
  • Divide both sides by \(-0.7\) to solve for \(w\): \(w = \frac{1.4}{-0.7}\).
  • Simplification results in \(w = -2\).
This step might involve arithmetic operations like division or multiplication to solve directly for the variable. The ultimate goal here is to arrive at a single clear value for the variable, completing the problem-solving process.

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Most popular questions from this chapter

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