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Chapter 2: Problem 71

Solve each equation, and check the solution. $$ 8 w-4 w+w=-3 $$

Short Answer

Expert verified
The solution is \(w = \frac{-3}{5}\).

Step by step solution

01

Combine Like Terms

First, combine the like terms on the left side of the equation: \(8w - 4w + w\).
02

Simplify the Expression

Simplify the expression by adding the coefficients of the variable \(w\): \(8w - 4w + w = (8-4+1)w = 5w\). So the equation becomes: \(5w = -3\).
03

Solve for w

To isolate \(w\), divide both sides of the equation by 5: \(w = \frac{-3}{5}\).
04

Check the Solution

Substitute \(w = \frac{-3}{5}\) back into the original equation to verify: \(8w - 4w + w = 8\left( \frac{-3}{5} \right) - 4\left( \frac{-3}{5} \right) + \left( \frac{-3}{5} \right)\). Calculate each term: \( 8\left( \frac{-3}{5} \right) = \frac{-24}{5}\) \( -4\left( \frac{-3}{5} \right) = \frac{12}{5}\) \( \left( \frac{-3}{5} \right) = \frac{-3}{5}\) Now combine: \( \frac{-24}{5} + \frac{12}{5} + \frac{-3}{5} = \frac{-24 + 12 - 3}{5} = \frac{-15}{5} = -3\). Since both sides are equal, the solution \(w = \frac{-3}{5}\) is verified.

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Key Concepts

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

Combining Like Terms
Combining like terms is an essential step in solving linear equations. Like terms are terms that have the same variable raised to the same power. In the equation given, we have:
\(8w - 4w + w\).
Notice that all terms have the variable \(w\). Hence, they are like terms.
  • Start by listing all the coefficients of \(w\): 8, -4, and 1.
  • Combine them by adding: \(8 - 4 + 1\)
  • This simplifies to: \(5w\)

Combining like terms simplifies the equation and makes it easier to solve. Now we have a much simpler equation:
\(5w = -3\)
Isolating Variables
To solve the equation \(5w = -3\), we need to isolate the variable \(w\).
Isolating the variable means getting it by itself on one side of the equation. Here鈥檚 how you can do it:
  • Look at the equation: \(5w = -3\)
  • Notice that \(w\) is multiplied by 5. To isolate \(w\), we need to undo this multiplication.

    • This is done by dividing both sides of the equation by 5:
    \[ \frac{5w}{5} = \frac{-3}{5} \]
    The left-hand side simplifies to \(w\), giving us:
    \[ w = \frac{-3}{5} \].
    Now, \(w\) is isolated, and we have the solution to the equation.
Verifying Solutions
After solving for \(w\), it's important to verify the solution to ensure accuracy.
We found \(w = \frac{-3}{5}\). To verify, substitute \(w\) back into the original equation:
\(8w - 4w + w = -3\) becomes:
\[ 8 \left( \frac{-3}{5} \right) - 4 \left( \frac{-3}{5} \right) + \left( \frac{-3}{5} \right) \]
Let's simplify each term:
  • \( 8 \left( \frac{-3}{5} \right) = \frac{-24}{5} \)
  • \( -4 \left( \frac{-3}{5} \right) = \frac{12}{5} \)

  • \( \left( \frac{-3}{5} \right) = \frac{-3}{5} \)

Combine the results:
\[ \frac{-24}{5} + \frac{12}{5} + \frac{-3}{5} = \frac{-24 + 12 - 3}{5} = \frac{-15}{5} = -3 \]
Since both sides of the equation are equal, our solution \(w = \frac{-3}{5}\) is verified as correct.

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