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When solving linear equations, expanding parentheses is a crucial step towards simplifying the equation. It involves distributing a factor into each term within a parenthesis. This is guided by the distributive property of multiplication over addition. For example, if you encounter a scenario where you have a number outside of the parentheses, as in the original exercise \(0.4(20)\), you would perform the multiplication \(0.4 \times 20\) to eliminate the parentheses.

For each term inside the parentheses, multiplication by the number outside is required. This process of expansion helps in breaking down the equation into more manageable pieces and reveals like terms, which can then be consolidated, moving us to the next step in solving the linear equation.

Once parentheses are expanded, the next step is to consolidate like terms. Like terms are terms that contain the same variable raised to the same power. In the equation \(0.7x + 8 = 0.5x + 10\), after expanding the parentheses, we have terms with the variable \(x\) on both sides of the equation. Consolidating like terms involves combining these terms to simplify the equation.

To consolidate, we'll subtract \(0.5x\) from both sides to collect the \(x\)-terms on one side of the equation, reducing it to \(0.2x = 2\). This makes it easier to see the relationship between the variable and its coefficient, setting the stage for isolating the variable—our ultimate goal in solving for \(x\).

The final stretch in solving a linear equation is isolating the variable. Our aim is to get the variable on one side of the equation by itself, clearly displaying its value. Once like terms are consolidated, as in \(0.2x = 2\), we can isolate the variable by performing the opposite operation of the coefficient next to the variable.

In this case, \(x\) is multiplied by \(0.2\), so we do the opposite—divide both sides of the equation by \(0.2\). This gives us \(x = \frac{2}{0.2}\), resulting in \(x = 10\). Remember, the division is the inverse operation of multiplication, and this act of balancing the equation by doing the same operation on both sides is what helps us isolate \(x\). Isolating the variable is the concluding step that provides us with the solution to the linear equation.