Problem 26 - Prealgebra

Question

Solve each equation. Check your solution. $$4.1 p=16.4$$

Step-by-Step Solution

Answer

The solution is \(p = 4\).

Step 1: Understand the Equation

We have the equation: \(4.1p = 16.4\). This means we need to find a value of \(p\) such that when multiplied by 4.1, it equals 16.4.

Step 2: Isolate the Variable

To solve for \(p\), we divide both sides of the equation by 4.1 to isolate \(p\). This gives us the equation: \(p = \frac{16.4}{4.1}\).

Step 3: Perform the Division

Perform the division \(\frac{16.4}{4.1}\). We can calculate this division to find the value of \(p\). Here, \(16.4 \div 4.1 = 4\), so \(p = 4\).

Step 4: Check the Solution

We check our solution by substituting \(p = 4\) back into the original equation \(4.1p = 16.4\). Substitute \(4\) for \(p\): \(4.1 \times 4 = 16.4\). This verifies our solution is correct, since the left side equals the right side.

Key Concepts

Prealgebra

Prealgebra is the foundation of all higher math skills, focusing on concepts like arithmetic, basic algebraic thinking, and manipulating simple equations. It's the stepping stone that prepares you for more complex topics. When we encounter an equation such as \(4.1p = 16.4\), we're using prealgebra skills to solve for an unknown variable.
In prealgebra:

• We often work with simple equations involving addition, subtraction, multiplication, and division.


• We use operations to balance equations and solve for unknowns.


• Understanding decimals and basic number theory is an essential part.

Mastering these basics ensures that you have a solid groundwork for tackling algebra and beyond.

Isolate the Variable

Isolating the variable is a core concept in algebra, especially when solving equations. It means rearranging the equation in such a way that the variable you are solving for stands alone on one side of the equation.

For example, in the equation \(4.1p = 16.4\), the goal is to get \(p\) by itself. To do this, you need to perform operations that simplify the equation step by step until \(p\) is isolated. Here's how:

• Step 1: Identify the operation applied to \(p\) (in this case, \(p\) is multiplied by 4.1).


• Step 2: Use the inverse operation to cancel out this multiplication. Since \(p\) is multiplied by 4.1, you would divide both sides by 4.1.


• Step 3: Simplify to find \(p = \frac{16.4}{4.1}\).

Once you have isolated \(p\), you can easily find its value. This systematic approach is crucial for successfully solving equations.

Checking Solutions

After solving an equation, it is important to check if the solution is correct. This ensures that no mistakes were made during the calculation process.

To check our solution for \(4.1p = 16.4\):

• Substitute the value you found for the variable back into the original equation. In this case, insert \(p = 4\) into the equation, making it \(4.1 \times 4\).


• Perform the arithmetic: calculate \(4.1 \times 4 = 16.4\).


• Verify: Check to see if the left side of the equation equals the right side, \(16.4\).

If both sides match, your solution is verified as correct. This step helps confirm your understanding and ensures the accuracy of your work.