Question

Solve \(p-5=12\)

Short answer

The value of \( p \) is 17.

Step-by-step solution

Identify the Equation

The given equation is: \[ p - 5 = 12 \]

Isolate the Variable

To isolate the variable \( p \), add 5 to both sides of the equation. This will help to cancel out the -5 on the left side. \[ p - 5 + 5 = 12 + 5 \]

Simplify

After adding 5 to both sides, simplify the expression. On the left side, the -5 and +5 cancel each other out, leaving only \( p \).\[ p = 12 + 5 \]Simplify the right side of the equation to get the final value of \( p \).\[ p = 17 \]

Key concepts

Isolating Variables

Isolating the variable means getting the variable by itself on one side of the equation. This is an essential step in solving linear equations. Consider our example equation, \( p-5=12 \).
To isolate \( p \), we need to move the -5 from the left side to the right. We do this by performing the opposite operation of subtraction, which is addition. In simpler terms, if something is subtracted on one side, we add it to both sides:
\( p-5 + 5 = 12 + 5 \).
By adding 5 to both sides, we successfully isolate \( p \) because -5 and +5 cancel each other out on the left side, leaving us with:
\( p = 12 + 5 \). This technique is essential in basic algebra and beyond. Think of it as undoing whatever is being done to the variable. If the variable is multiplied by a number, you would divide both sides by that number to isolate it.

Simplification

Simplification is the next key step after isolating the variable. This involves performing any arithmetic operations to simplify the equation. In our example, after isolating \( p \), we end up with: \( p = 12 + 5 \)
The next step is to simplify the right side by actually adding 12 and 5 together. Simplifying means making the equation as easy to understand as possible, often involving basic arithmetic operations like addition, subtraction, multiplication, or division. Adding 12 and 5 gives us: \( p = 17 \)
So the fully simplified and isolated solution is \( p = 17 \).
Simplification is crucial because it gives you the most reduced form of your answer, which is easier to understand and check.

Basic Algebra

Basic algebra involves understanding and manipulating mathematical symbols and variables to solve equations. Being comfortable with concepts like variables, constants, and arithmetic operations lays the foundation. Let's quickly recap our example:

• We started with the equation \( p - 5 = 12 \).
• We isolated the variable \( p \) by adding 5 to both sides.
• Finally, we simplified the equation to find \( p = 17 \).

In essence, basic algebra requires you to systematically follow rules and operations that will reveal the value of the unknown variable. Performing operations like adding, subtracting, multiplying, or dividing both sides of an equation helps to maintain the equality and ultimately solves the problem.