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Understanding the distributive property is a fundamental skill in algebra. It allows you to multiply a single term by each term within a parenthesis. For example, in the given problem where we have \(1.5(x+8)\), we distribute \(1.5\) by multiplying it with \(x\) and then with \(8\). This action simplifies the expression to \(1.5x + 12\).

The beauty of the distributive property is that it works in reverse too. If we want to factor out a common term from several terms, we can do so. This property lays the foundation for solving more complex equations and is used in conjunction with other properties to simplify and solve algebraic expressions.

Moving on to combining like terms, which is another way to simplify equations. Like terms are terms that have the exact same variables raised to the same power. They can be combined by adding or subtracting their coefficients. In the context of our problem, after distributing, we have terms with the variable \(x\) on both sides of the equation: \(1.5x + 12 = 3.2 + 0.7x\).

To simplify this, we subtract \(0.7x\) from both sides which effectively combines the \(x\) terms together making it possible to gather them on one side of the equation. This step is critical to getting to a point where the variable we are solving for is isolated, which is the next crucial step in solving linear equations.

The last key step in solving linear equations is the isolation of variables. This means manipulating the equation in such a way that you get the variable you're solving for by itself on one side of the equation. After combining like terms, we obtained \(0.8x + 12 = 3.2\). From here, we need to isolate \(x\), which we do by moving the constant term to the other side of the equation through subtraction or addition. In our example, subtracting \(12\) from both sides gave us \(0.8x = -8.8\).

Lastly, we divide both sides by \(0.8\) to completely isolate \(x\). The final step not only leaves \(x\) on its own but also gives us the solution to the equation. Remember to always check your solution by substituting it back into the original equation. If both sides of the equation remain equal, you've found the correct solution!