91Ó°ÊÓ

Isolating variables is a fundamental method in solving linear equations. It involves manipulating an equation in such a way that the variable of interest, usually represented by a letter such as 'x', is on one side of the equation and all the numbers or constants are on the other side. When you isolate the variable, you can clearly see its value without any other terms interfering.

For instance, in the given exercise \( -7 = x - 15 \), the goal is to isolate \( x \) on one side. To achieve this, you perform the same mathematical operation on both sides of the equation, maintaining the balance and integrity of the equation. By adding 15 to both sides, we can move all constant numeric values away from the variable, hence isolating \( x \).

This process is instrumental as it simplifies complex equations into an easily understandable form where the value of the variable is clearly presented as \( x = 8 \) in our example.

The addition property of equality is an essential principle in solving equations. It asserts that if you add the same number to both sides of an equation, the equality is still valid. This concept is pivotal when adjusting an equation to isolate a variable.

In our exercise, we apply this property by adding 15 to both sides of the initial equation \( -7 = x - 15 \). As a result of this action, we obtain \( -7 + 15 = x - 15 + 15 \). Simplifying this gives us the new, balanced equation \( 8 = x \). The understanding and correct application of this property ensures that the solutions we find are consistent with the original equation.

Once a solution to an equation has been found, it is crucial to verify its correctness. Checking solutions helps to confirm that our mathematical manipulations have preserved the equation’s integrity and that the found value for the variable indeed satisfies the original equation.

To check the solution \( x = 8 \) from our example, we substitute \( x \) with the value 8 back into the original equation \( -7 = x - 15 \). If our solution is correct, the equation will hold true. Upon substitution, we get \( -7 = 8 - 15 \), which simplifies to \( -7 = -7 \). Both sides of the equation are equal, verifying that the solution is indeed correct. This crucial step should never be skipped, as it not only confirms the solution but also builds confidence in understanding the solving process.