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When solving linear equations, understanding 'like terms' is essential. Like terms are terms that have the same variables raised to the same power. In the equation provided, 7.7r and 6.7r are like terms because they both contain the variable r.

Like terms can be combined to simplify equations. For instance, in the equation 7.7r - 6 = 6.7r - 6, both sides can be simplified by subtracting 6.7r. This helps reduce complexity and makes it easier to isolate the variable.

By identifying and combining like terms, equations become simpler and more manageable, allowing us to focus on the main goal: isolating the variable. This is the key to moving forward in solving any linear equation.

Isolating the variable is a critical step in solving linear equations. The goal is to get the variable on one side of the equation by itself. Let's look at how this is done in our example: Starting with 7.7r - 6 = 6.7r - 6.

Our first step was to subtract 6.7r from both sides, resulting in 1r - 6 = -6. This helped to get terms with the variable r on one side. The next step is to eliminate any constants on the side with the variable. We achieve this by adding 6 to both sides: r - 6 + 6 = -6 + 6. This leaves us with r = 0.

Isolating the variable becomes straightforward when we perform the same operation (addition, subtraction, etc.) on both sides of the equation. This preserves equality and helps us solve for the variable.

Algebraic manipulation involves using mathematical operations to simplify and solve equations. In our example, we used subtraction and addition to manipulate the equation effectively.

First, by subtracting 6.7r from both sides, we combined like terms, simplifying the equation to 1r - 6 = -6. Next, we added 6 to both sides to isolate the variable r, resulting in r = 0.

Understanding how to use these basic operations is crucial for solving equations. These processes are not just limited to simple equations but are applicable in more complex algebraic problems as well.

Mastering algebraic manipulation provides a strong foundation for more advanced mathematics. With practice, these steps become intuitive and make solving equations a straightforward process.