91Ó°ÊÓ

When solving linear equations, one of the key steps is to isolate the variable. Isolating the variable means getting the variable by itself on one side of the equation. This allows us to find its value. In the equation given:

\(x - 18 = 22\), our variable is \(x\). To isolate \(x\), you need to eliminate the constant that is being subtracted from \(x\).

This is done by performing the opposite operation. Since 18 is being subtracted, we need to add 18 to both sides of the equation:

\(x - 18 + 18 = 22 + 18\).
By adding 18 to both sides, we balance the equation and successfully isolate the variable. Now, the equation simplifies to:
\(x = 40\).

Simplifying equations is a crucial step in solving them efficiently. Once you have manipulated the equation to isolate the variable, the next task is to simplify both sides. This means performing any arithmetic operations to get the simplest form of the equation. In our example:

\(x - 18 + 18 = 22 + 18\), we performed the addition on both sides to get:
\(x = 40\)

In this case, 'simplifying' was straightforward. However, in more complicated equations, this could involve combining like terms, reducing fractions, or applying distributive properties.

Regardless of the complexity, the goal remains the same: to make the equation as simple as possible so that the value of the variable is clear and easy to understand.

The step-by-step solution is essential in breaking down the process to solve linear equations. Let's go through the steps using our example:

1. **Understand the Equation**: First, recognize what the equation represents and identify your variable and constants. For our exercise, the equation is \(x - 18 = 22\).

2. **Isolate the Variable**: Use inverse operations to get the variable alone. Our goal is to find the value of \(x\). Adding 18 to both sides gets us: \(x - 18 + 18 = 22 + 18\).
By doing so, we eliminate the -18 from the left side.

3. **Simplify Both Sides**: Perform the arithmetic operations on both sides to simplify the equation. After adding 18, we get: \(x = 40\).
Now we have the simplified form and the variable is isolated.

These steps make solving equations systematic. By understanding and applying each step, you can tackle any linear equation confidently and correctly.