91Ó°ÊÓ

The primary goal when solving linear equations is often to find the value of the variable, which in our original problem is represented by \( x \). This involves a method called 'isolation of variables'. This means that you want to have your variable (here, \( x \)) alone on one side of the equation. In simpler terms, you want all other numbers and terms on the opposite side.
Let's look at the original equation: \( x - 6 = 1 \). To isolate \( x \), we need to "undo" the -6. We do this by performing the opposite operation of subtraction, which is addition.

• Add 6 to both sides of the equation.
• By doing this, the -6 on the left cancels out, leaving the variable \( x \) by itself.

This will lead us to the isolated variable, allowing us to identify its value given the remaining side of the equation.

After successfully isolating a variable, the next crucial step in solving linear equations is equation simplification. This process involves making both sides of the equation as straightforward as possible. Let's see how this works with our equation.
We have just isolated \( x \) by adding 6, resulting in the equation \( x = 1 + 6 \). Simplifying this equation means performing the arithmetic on the right-hand side. This is a simple operation since it involves the straightforward addition of numbers:

• The left side is already simplified to \( x \) because \( -6 + 6 = 0 \).
• On the right, you simplify \( 1 + 6 \) which equals 7.

Therefore, the simplified equation leaves us with \( x = 7 \), which is the solution. Simplification is essential as it not only helps in finding the solution but also ensures clarity and understanding of the problem.

At the core of solving linear equations, like the one given, are basic arithmetic operations. These include addition, subtraction, multiplication, and division. Understanding and effectively applying these operations can make solving equations much easier. In our exercise:

• Subtraction is the starting point where 6 is subtracted from \( x \).
• Addition is then used to reverse this subtraction when we add 6 to both sides. This operation directly contributed to isolating the variable.

Recognizing and applying these operations with precision is key to correctly solving the equation. In our example, using subtraction and addition allowed us to find the simple solution \( x = 7 \). This demonstrates the power of basic arithmetic in untangling more complex mathematical problems.