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Chapter 2: Problem 12

Solve each equation, and check your solution. $$ -7 w-4=10 $$

Short Answer

Expert verified
The solution is $$ w = -2 $$. Verification shows it is correct.

Step by step solution

01

- Isolate the term with the variable

Start by isolating the term with the variable on one side of the equation. To do this, add 4 to both sides of the equation: $$ -7w - 4 + 4 = 10 + 4 $$ This simplifies to: $$ -7w = 14 $$
02

- Solve for the variable

Next, solve for the variable by dividing both sides of the equation by -7: $$ w = \frac{14}{-7} $$ This simplifies to: $$ w = -2 $$
03

- Check the solution

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

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Key Concepts

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

Isolate the Variable
The first step in solving a linear equation is to isolate the variable. In the given equation, \( -7w - 4 = 10 \), our goal is to get the variable \( w \) on one side of the equation and the constants on the other side. To start, we perform operations that will isolate \( -7w \).

You can follow these strategies:
  • Add or subtract terms to both sides of the equation to move constants away from the variable term.
  • In this case, adding 4 to both sides removes the -4 on the left side:
<\( -7w - 4 + 4 = 10 + 4 \)
This simplifies to:
\( -7w = 14 \)
This leaves just the variable term \( -7w \) on the left side.

Remember, whatever you do to one side of the equation, you must do to the other to maintain balance.
Inverse Operations
Once the variable term is isolated, the next step is to solve for the variable using inverse operations. In our example, we have:<\( -7w = 14 \)
Here, \( -7 \) is multiplied by \( w \).

To remove the multiplication, we do the opposite, which is division. You divide both sides by -7:
\( w = \frac{14}{-7} \)
This simplifies to:
\( w = -2 \)
Inverse operations are essentially doing the opposite mathematical operation to both sides to isolate the variable.

Some common inverse operations include:
  • Addition vs. Subtraction
  • Multiplication vs. Division
  • Exponential functions vs. Logarithms
Make it a habit to check which operation will help you simplify toward isolating the variable.
Checking Solutions
After solving for the variable, it is crucial to check your solution to ensure it is correct. This can be done by substituting the found value back into the original equation. If both sides of the original equation are equal after substitution, the solution is verified as correct.

In our equation:\( -7w - 4 = 10 \)
we found \( w = -2 \). Substitute -2 back into the original equation:
\( -7(-2) - 4 = 10 \)
Simplify inside the parentheses:
\( 14 - 4 = 10 \)
As both sides are equal, this confirms \( w = -2 \) is the correct solution.

Checking solutions is important for:
  • Validating the accuracy of your calculations
  • Catching any potential mistakes
  • Gaining confidence in your problem-solving skills
Never skip this final step; it provides an essential verification of your work.

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Most popular questions from this chapter

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