Problem 69 - Elementary Algebra

Question

Solve the equation \(y-2=-2\).

Step-by-Step Solution

Answer

Answer: The value of y that satisfies the equation is y = 0.

Step 1: Identify the given equation

The given equation is y - 2 = -2.

Step 2: Isolate the variable y

To isolate y, we need to get rid of the term "-2" from the left side of the equation. We do this by adding 2 to both sides of the equation: y - 2 + 2 = -2 + 2

Step 3: Simplify the equation

Now, we can simplify the equation by canceling out -2 and 2 on the left side and adding -2 and 2 on the right side: y = 0

Step 4: State the solution

The solution to the equation y - 2 = -2 is y = 0.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics, particularly when it comes to linear equations such as the one given here: \(y - 2 = -2\). When solving an equation, the goal is to find the value of the unknown variable that makes the equation true. It involves manipulating the equation in such a way that the variable is left alone on one side of the equation, with its corresponding value on the other side.

There are a few key steps that help in solving equations efficiently:

• Understand what is being asked: Identify the equation and the variable you need to solve for.
• Perform operations: Use basic mathematical operations like addition, subtraction, multiplication, or division to rearrange the equation.
• Check your solution: Substitute your answer back into the original equation to ensure that it satisfies the equation.

By following these steps, you can develop a systematic approach to solving equations, making it easier each time you encounter a new problem.

Isolate Variable

To solve an equation effectively, isolating the variable is often one of the first and most important steps. In the given exercise, the variable to isolate is \(y\), and isolating the variable means we want \(y\) by itself on one side of the equation.

Let's take a closer look at the original equation: \(y - 2 = -2\). Here’s how you would isolate \(y\):

• Add or subtract terms: In this example, we need to eliminate \(-2\) from the left side. We do this by adding \(+2\) to both sides, effectively canceling out the \(-2\).
• Maintain the balance: Whatever operation is performed on one side of the equation must be done to the other side. This ensures that the equation remains balanced.

After performing these steps, the equation simplifies to \(y = 0\), where \(y\) is isolated, giving us the solution. This process is crucial because it prepares the equation for the final step of finding the value of the variable.

Basic Algebra

Basic algebra is the foundation of solving equations like \(y - 2 = -2\). At its core, algebra deals with finding unknown values and understanding the relationships between numbers using letters or symbols to represent these values.

Key concepts in basic algebra include:

• Variables: Letters such as \(y\) that stand in place of unknown numbers.
• Constants: Numbers that are fixed or known values, like the \(-2\) in our equation.
• Operations: Basic mathematical operations (addition, subtraction) that are used to manipulate the equation.

By using these concepts, algebra helps in creating a straightforward method to find unknowns, demonstrating the power of applying operations logically and systematically. This piece of algebra not only aids in problem-solving tasks but also enhances logical thinking and reasoning skills.