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Chapter 3: Problem 44

Solve the equation. $$p-6=-18$$

Short Answer

Expert verified
The solution is \( p = -12 \).

Step by step solution

01

Identify the equation

The given equation is \( p - 6 = -18 \).
02

Isolate the variable

To isolate \( p \), add 6 to both sides of the equation: \( p - 6 + 6 = -18 + 6 \).
03

Simplify both sides

On the left side, \( -6 + 6 \) cancels out, leaving \( p \). On the right side, simplify \( -18 + 6 \) to get \( -12 \).
04

Write the solution

The simplified equation is \( p = -12 \).

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Key Concepts

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

Isolate the Variable
When solving linear equations, our goal is to find the value of the variable. In this case, our variable is 'p'. The first step is to isolate the variable, meaning we want 'p' to be alone on one side of the equation. To do this, we perform operations that will simplify our equation.

In the given equation, \( p - 6 = -18 \), 'p' is being subtracted by 6. To isolate 'p', we need to undo this operation. We can do this by adding 6 to both sides of the equation. This step cancels out the -6 on the left side, leaving 'p' by itself. Remember: whatever you do to one side of the equation, you must do to the other side as well!

This step looks like this:
\( p - 6 + 6 = -18 + 6 \)
Simplification
After performing operations to isolate the variable, we simplify the equation. Simplification involves combining like terms and performing arithmetic operations. In our equation, we added 6 to both sides to get:

\( p - 6 + 6 = -18 + 6 \)

Now, we simplify each side of the equation separately. On the left side, \( p - 6 + 6 \) simplifies to 'p' because \( -6 + 6 \) equals 0. This leaves us with:

\( p = -18 + 6 \)

On the right side, \( -18 + 6 \) simplifies to -12. Therefore, we end up with:

\( p = -12 \)
Solving Linear Equations
Solving linear equations is a foundational skill in algebra. A linear equation is any equation that can be written in the form \( ax + b = c \). The goal is to find the value of 'x' (or any variable) that makes the equation true.

Here's a step-by-step breakdown for solving linear equations:
  • Identify the equation.
  • Isolate the variable by performing inverse operations (addition or subtraction).
  • Simplify the equation by combining like terms.
  • Write the solution once the variable is isolated.
Let's revisit our example, \( p - 6 = -18 \), and see these steps in action:

Starting with the equation, isolate 'p' by adding 6 to both sides:

\( p - 6 + 6 = -18 + 6 \)

Then, simplify both sides to get:

\( p = -12 \)

Thus, the solution is \( p = -12 \). Now you know the value of 'p' that satisfies the original equation.

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