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Chapter 2: Problem 10

Solve each equation, and check the solution. \(-7 w-4=10\)

Short Answer

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

Step by step solution

01

Add 4 to both sides

To isolate the term with the variable, add 4 to both sides of the equation:\[ -7w - 4 + 4 = 10 + 4 \]This simplifies to:\[ -7w = 14 \]
02

Divide by -7

Next, divide both sides of the equation by -7 to solve for \( w \):\[ \frac{-7w}{-7} = \frac{14}{-7} \]This simplifies to:\[ w = -2 \]
03

Check the solution

Substitute \( w = -2 \) back into the original equation to ensure it is correct:\[ -7(-2) - 4 = 10 \]Simplify inside the parentheses:\[ 14 - 4 = 10 \]Since both sides of the equation are equal, the solution \( w = -2 \) is verified.

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Key Concepts

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

Adding and Subtracting Terms
When solving linear equations, it's common to use addition or subtraction to simplify the equation. The goal is to move terms from one side of the equation to the other to isolate the variable. In our example, the equation is \( -7w - 4 = 10 \).
  • First, identify the term you need to eliminate to simplify the equation. Here, it鈥檚 the \( -4 \) on the left side.
  • To get rid of \( -4 \), add 4 to both sides of the equation. This keeps the equation balanced:
    • \[ -7w - 4 + 4 = 10 + 4 \]
  • This simplifies to:
    • \[ -7w = 14 \]
  • By adding 4 to both sides, you maintain equality and simplify the equation, now focuses solely on the term with the variable.
Isolating the Variable
One of the main goals in solving linear equations is isolating the variable. This means getting the variable on one side of the equation by itself. In the simplified equation \(-7w = 14\):
  • Notice the variable \(w\) is multiplied by \(-7\).
  • To isolate \(w\), we need to undo the multiplication. This is done in the next section by division.
  • Remember, when isolating the variable, every step must maintain the equation's balance. Whatever you do to one side, you must do to the other.
Successfully isolating the variable turns the equation into a simpler form, making it easier to solve.
Division in Equations
Dividing both sides by the same number is the last step in solving the linear equation. From the simplified equation \(-7w = 14\):
  • We need to divide both sides by \(-7\) to isolate \(w\):
    • \[ \frac{-7w}{-7} = \frac{14}{-7} \]
  • This simplifies to:
    • \[ w = -2 \]
  • Dividing by the same number (in this case \(-7\)) ensures that the equality is maintained. Thus, we find that \(w = -2\).
  • After solving for \(w\), it's important to check the solution by substituting the value back into the original equation.
This confirms our solution is correct and reinforces the steps taken to isolate and solve for the variable.

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Most popular questions from this chapter

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