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The distributive property is a fundamental algebraic principle used to simplify expressions. It allows you to multiply a single term by each term inside a set of parentheses. This comes in handy when dealing with equations like the one given: \[ 6x - 4(-3x + 2) = 10 \].
In this example, the distributive property enables us to eliminate the parentheses:

• Multiply \(-4\) by \(-3x\) to get \(+12x\).
• Multiply \(-4\) by \(2\) to get \(-8\).

So, the expression \(-4(-3x + 2)\) becomes \(+12x - 8\). When we apply this to the equation it transforms to: \[ 6x + 12x - 8 = 10 \].
By using the distributive property, we've simplified the equation, making it easier to solve. This property is essential when working with algebraic expressions, particularly when they get complex with multiple terms involved.

Once we have applied the distributive property and simplified the equation, the next step is to combine like terms. 'Like terms' refer to terms that have the same variables raised to the same power. In the equation \[ 6x + 12x - 8 = 10 \], the terms \(6x\) and \(12x\) are considered like terms because both involve the variable \(x\).
When combining like terms, you simply add or subtract their coefficients:

• Add \6x\ and \12x\ together to get \18x\.

Thus, the equation simplifies to: \[ 18x - 8 = 10 \]. The process of combining like terms condenses the equation further, which is essential before we can solve for the variable.
Finally, we isolate \(x\) by moving other terms across the equation, making it easy to find the solution.

After obtaining a solution, verifying its accuracy is crucial. Checking your solutions ensures that no mistakes were made during the simplification process and that your result satisfies the original equation.
In this exercise, we found that \(x = 1\). To check this:

• Substitute \(x = 1\) back into the original equation: \[ 6(1) - 4(-3(1) + 2) = 10 \].
• Simplify: \(6 + 4(3 - 2)\).
• This further simplifies to \(6 + 4\), which equals \10\.

Since both sides of the equation are equal, \(10 = 10\), it confirms that \(x = 1\) is indeed the correct solution.
Checking solutions is a valuable practice as it provides confidence that your calculations are correct, securing your understanding and application of the process.