91Ó°ÊÓ

Understanding the distributive property is crucial when solving linear equations involving parentheses. It allows us to remove the parentheses by distributing a multiplication over addition or subtraction within the parentheses. In the context of our exercise, \(12x - 2(4x - 6) = 28\), we need to multiply -2 by each term inside the parentheses. This operation transforms \( -2*(4x) \) to \( -8x \) and \( -2*(-6) \) to \( +12 \), thus eliminating the parentheses and making the equation simpler.

When applying the distributive property, always ensure you multiply the term outside the parentheses with every term inside, paying close attention to the sign (positive or negative) of each term. It's a systematic technique that simplifies the equation, setting the stage for the next steps.

Once we have used the distributive property, the next step is combining like terms. In our example, after distribution, we obtained the equation \(12x - 8x + 12 = 28\). Here, the 'like terms' are those that contain the same variable raised to the same power—in this case, \(12x\) and \( -8x\).

To combine them, you simply add or subtract their coefficients, which are the numbers in front of the variables. So, \(12x - 8x\) becomes \(4x\), leaving us with the easier equation \(4x + 12 = 28\). It's essential to recognize like terms to simplify the equation as much as possible, which in turn, aids in solving for the variable more efficiently.

Isolating the variable is the final and most critical step in solving linear equations. It involves performing operations that will 'free' the variable from other numbers or terms. In our exercise, after combining like terms, we're left with \(4x + 12 = 28\). Now, to isolate \(x\), we need to get rid of the \(+12\) by performing the opposite operation—subtraction—and thus we subtract 12 from both sides of the equation.

This step shows us that \(4x = 16\). We're close to isolating \(x\) completely; the last move is to divide both sides by the coefficient of \(x\), which is 4. Doing this, the equation simplifies to \(x = 4\), giving us the solution. Remember, balancing the equation by performing the same operation on each side is paramount in maintaining the equation's integrity.