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In solving linear equations, one of the first steps often involves isolating the variable of interest. This means rearranging the equation so that the variable appears on one side while everything else is on the opposite side. This helps simplify the equation and makes solving it more straightforward.

In the equation provided, \(7x - 12 = 3x\), the aim is to collect all the \(x\) terms on one side. To do this, you subtract \(3x\) from both sides. This action doesn't change the equation’s balance since you’re performing the same operation on both sides. As a result of this, you get \(7x - 3x = 12\), simplifying to \(4x = 12\). You now have the variable \(x\) isolated to one side, making it easier to solve.

After isolating the variable and finding its value, it's essential to confirm that your solution is correct. The substitution method is useful for this verification. It involves replacing the variable with the found solution in the original equation to ensure both sides are equal.

For the equation \(7x - 12 = 3x\), once we've solved for \(x\) and determined \(x = 3\), we substitute this value back into the original equation. Replacing \(x\) with \(3\), the equation becomes \(7 \times 3 - 12 = 3 \times 3\). Calculate each side separately:

• Left side: \(7 \times 3 - 12 = 21 - 12 = 9\)
• Right side: \(3 \times 3 = 9\)

Since both sides equal \(9\), the solution \(x = 3\) is confirmed to be correct.

Checking the results of your calculations is a crucial step in solving equations. This mathematical verification ensures the solution satisfies the original equation. This practice helps avoid errors and reinforces your understanding of the solution.

In the example equation \(7x - 12 = 3x\), once you find \(x = 3\), it’s verified through substitution. This means placing \(x = 3\) back into the equation to ensure both sides are equivalent. We've calculated that substituting \(3\) yields equal results on both sides: \(21 - 12 = 9\) (left side) and \(9\) (right side).

• If both sides match, as they do here, it verifies that the solution is correct and accurate.
• This step helps prevent mistakes in more complex problems too, aiding in developing thorough problem-solving skills.

With this confirmation, you can confidently state that the derived solution is the correct one.