Problem 202 Solve the following equations wi... [FREE SOLUTION]

Chapter 2: Problem 202

Solve the following equations with variables and constants on both sides. $$2 p-1=4 p-33$$

Short Answer

Expert verified

p = 16

Step by step solution

- Move Variables to One Side

Subtract $2p$ from both sides to get all the variable terms on one side: $2p - 1 - 2p = 4p - 33 - 2p$ which simplifies to: $-1 = 2p - 33$

- Move Constants to One Side

Add $33$ to both sides to isolate the variable term: $-1 + 33 = 2p - 33 + 33$ This simplifies to: $32 = 2p$

- Solve for the Variable

Divide both sides by $2$ to solve for $p$: $32 / 2 = 2p / 2$ which results in: $p = 16$

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Key Concepts

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

Variables and Constants

In any equation, you will often see letters and numbers. The letters are called **variables** and they represent unknown values. In the equation, they are usually denoted by symbols like $x$, $p$, or $y$. On the other hand, numbers are called **constants** because their values do not change. For instance, in the equation $2p - 1 = 4p - 33$, $p$ is the variable, while $2$, $1$, $4$, and $33$ are constants. Understanding these terms is crucial, as they form the foundation of algebraic equations. Moving forward, you will learn how to manipulate them to find the solution to the exercise.

Isolating the Variable

To solve an equation, your goal is to find the value of the variable. This requires isolating the variable on one side of the equation. Let's take the equation $2p - 1 = 4p - 33$, as an example. In this step, we start by moving all terms involving the variable to one side. We can achieve this by subtracting $2p$ from both sides: $2p - 1 - 2p = 4p - 33 - 2p$. This simplification leaves us with $-1 = 2p - 33$. Now, the variable term $2p$ is isolated on one side of the equation.

Algebraic Manipulation

Once the variable is isolated, the next step involves algebraic manipulation to solve for the variable. We continue with our equation $-1 = 2p - 33$. The goal here is to isolate $p$. To do this, we need to move the constant term $-33$ to the other side by adding $33$ to both sides of the equation: $-1 + 33 = 2p - 33 + 33$. With simplification, we get $32 = 2p$. The final step is to divide both sides by $2$ to solve for \p\: $32 / 2 = 2p / 2$, which simplifies to $p = 16$. This process of rearranging, adding, and dividing is called algebraic manipulation. It is a crucial skill to solve any linear equations effectively.

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91影视

Solve the following equations with variables and constants on both sides. $$2 p-1=4 p-33$$

Short Answer

Step by step solution

- Move Variables to One Side

- Move Constants to One Side

- Solve for the Variable

Key Concepts

Variables and Constants

Isolating the Variable

Algebraic Manipulation

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