Problem 56 - Algebra A Combined Function

Question

\(2-0.4 p=2\)

Step-by-Step Solution

Answer

The solution is \(p = 0\).

Step 1: Understand the Equation

The given equation is \(2 - 0.4p = 2\). It is a linear equation that needs to be solved for the variable \(p\). Our goal is to isolate \(p\) by performing appropriate algebraic operations.

Step 2: Subtract 2 from Both Sides

To simplify the equation, subtract 2 from both sides to eliminate the constant term on the left. This gives:\[ 2 - 0.4p - 2 = 2 - 2 \]Simplifying both sides results in:\[ -0.4p = 0 \]

Step 3: Solve for p

To find the value of \(p\), divide both sides of the equation by \(-0.4\) to isolate \(p\):\[ p = \frac{0}{-0.4} \]Simplifying this gives:\[ p = 0 \]

Key Concepts

Algebraic Operations

Algebraic operations are the foundation of solving equations. They include addition, subtraction, multiplication, and division applied to both numbers and variables. In an equation like the one provided, we use these operations to transform and manipulate the expressions on both sides to isolate the variable. Think of it as the same operations you would use in arithmetic applied to unknowns as well.
- Subtracting numbers from both sides can help eliminate constants.
- Dividing by the coefficient of a variable is crucial for isolating it.
Using algebraic operations correctly ensures you maintain the balance of the equation, leading to the correct solution.

Isolate Variable

Isolating the variable is the key objective in solving a linear equation. This process involves performing algebraic operations to "free" the variable from the rest of the equation. Essentially, you want the variable on one side and everything else on the opposite.
In our example, this was done by first removing the constant term from the side of the variable by subtraction and then eliminating any coefficients through division.
Bearing in mind that every step should preserve the equation's balance, it's crucial to apply changes equally to both sides. Once isolated, you can easily determine the variable's value.

Simplifying Equations

Simplifying equations means breaking them down into simpler forms, making them easier to solve. This process involves combining like terms and performing the arithmetic necessary to tidy up the equation.
In the problem, simplifying began with reducing the equation after subtracting equal quantities from both sides, resulting in a simpler form: \[ -0.4p = 0 \]
- When you encounter terms and variables that can be cancelled or reduced, simplifying becomes a straightforward way to see the core of the problem.
- Being clear and methodical in simplifying ensures the solution is solved efficiently and accurately.
By simplifying effectively, you make complex equations much more manageable.