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Chapter 2: Problem 13

Solve each equation, and check your solution. $$ 4 p-5=2 p $$

Short Answer

Expert verified
The solution is \(p = \frac{5}{2}\) and it checks out correctly.

Step by step solution

01

- Isolate the variable term

Start by isolating the term with the variable on one side of the equation. Subtract \(2p\) from both sides: \(4p - 5 - 2p = 2p - 2p\). This simplifies to \(2p - 5 = 0\).
02

- Solve for the variable

Next, solve for \(p\) by adding 5 to both sides of the equation: \(2p - 5 + 5 = 0 + 5\). This simplifies to \(2p = 5\). Then divide both sides by 2: \(p = \frac{5}{2}\).
03

- Check the solution

Substitute the solution back into the original equation to check it. The original equation is \(4p - 5 = 2p\). Substitute \(p = \frac{5}{2}\): \(4 \times \frac{5}{2} - 5 = 2 \times \frac{5}{2}\). Simplify: \(10 - 5 = 5\) and \(5 = 5\), confirming the solution is correct.

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Key Concepts

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

Isolating Variables
Isolating the variable is the first step in solving linear equations. The goal is to get the variable by itself on one side of the equation. To do this, you need to perform operations that will move other terms to the opposite side.
For example, in the equation \(4p - 5 = 2p\), we want to isolate \(p\).
  • First, subtract the term with the variable on the other side, so subtract \(2p\) from both sides: \(4p - 2p - 5 = 2p - 2p\).
  • This simplifies to \(2p - 5 = 0\). Now, \(p\) is closer to being isolated.

The key idea here is to work towards having just one \(p\) on one side by performing inverse operations like addition, subtraction, multiplication, or division.
Always remember, whatever you do to one side of the equation, you must do to the other to keep it balanced.
Algebraic Manipulation
Algebraic manipulation involves using algebraic rules to simplify and solve equations. These rules include operations like addition, subtraction, multiplication, and division.
In our example equation \(2p - 5 = 0\), we need to solve for \(p\).
  • First, add 5 to both sides: \(2p - 5 + 5 = 0 + 5\).
  • This simplifies to \(2p = 5\).
  • Then, divide both sides by 2 to solve for \(p\): \(2p/2 = 5/2\).
  • This results in \(p = 5/2\).

Each step involves a straightforward application of basic algebraic principles. By consistently applying these rules, you can manipulate any linear equation to find the solution.

Practicing these manipulations will enhance your ability to handle more complex algebraic equations effortlessly.
Checking Solutions
After solving an equation, it is essential to verify that your solution is correct. This is done by substituting the solution back into the original equation.
Let's check our solution \(p = 5/2\) in the original equation \(4p - 5 = 2p\):
  • Substitute \(p = 5/2\) into the equation: \(4 \times 5/2 - 5 = 2 \times 5/2\).
  • Simplify both sides: \(10 - 5 = 5\) and \(5 = 5\).
Since both sides of the equation are equal, our solution is correct.

Checking solutions ensures that you have not made any mistakes during algebraic manipulation. It's a good habit to always double-check your answers to confirm their accuracy.

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Most popular questions from this chapter

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