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Chapter 2: Problem 7

Solve each linear equation. $$ -(w-12)=30 $$

Short Answer

Expert verified
w=-18.

Step by step solution

01

Distribute the Negative Sign

Start by distributing the negative sign across the parentheses: . This gives us .
02

Simplify the Expression

Now, simplify the left-hand side: . This leaves us with .
03

Solve for w

To isolate w, add 12 to both sides of the equation: . Simplifying this will give the solution to the equation.

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Key Concepts

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

Distributive Property
In this problem, we use the distributive property to eliminate the parentheses. The distributive property states that for any expressions \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true. In our exercise, we have a negative sign outside the parentheses, which can be considered as multiplying by \(-1\). To apply this:
  • Distribute the \(-1\) to both terms inside the parentheses:
    \(-1(w-12) = -1 \times w + -1 \times -12\)
  • This simplifies to \(-w + 12\).
Simplifying Expressions
Simplifying expressions is an important step in solving linear equations. After using the distributive property, the next step is to simplify further to make the equation cleaner and easier to work with. In our case, after distributing the negative sign:
  • We obtained \(-w + 12 = 30\).
  • At this point, make sure all like terms are combined if necessary. Here, there are no further like terms to combine, so the simplified equation remains as it is.
Isolating Variables
The ultimate goal is to solve for the variable \(w\). To achieve this, we need to isolate \(w\) on one side of the equation. Here鈥檚 how we do it:
  • We have \(-w + 12 = 30\).
  • Subtract 12 from both sides, resulting in \(-w = 30 - 12\).
  • This simplifies to \(-w = 18\).
Finally, to solve for \(w\), multiply both sides of the equation by \(-1\) to get:
  • \(w = -18\).

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