91Ó°ÊÓ

In mathematics, isolating the variable means rearranging an equation so that the variable of interest, often denoted as x, is by itself on one side of the equation. This practice is crucial for solving equations and determining the value of the unknown.

Take the given equation \(-5 = x + 8\). Here, we want to isolate the variable x. To do this, we need to remove the constants or any other non-variable terms that are on the same side as x.

In our equation, x is accompanied by a +8. We can isolate x by performing the opposite operation of addition, which is subtraction. Hence, we subtract 8 from both sides of the equation. This action ensures we maintain the equality and keep the equation balanced.

Solving equations involves finding the value of the unknown variable that makes the equation true. It's a fundamental skill in algebra.

Let’s apply the addition principle to solve the equation \(-5 = x + 8\). The addition principle states that you can add or subtract the same number from both sides of an equation without changing its equality. By subtracting 8 from both sides, you follow this principle:

\(-5 - 8 = x + 8 - 8\)

Simplifying this gives us:

\(-13 = x\)

Now, we have a solution for x: \(x = -13\). The equation is solved once we have isolated the variable on one side and simplified it fully.

To ensure the solution to an equation is correct, verification is essential. Verification involves substituting the solution back into the original equation and checking if the equality holds true.

Let's verify our solution \(x = -13\) with the original equation \(-5 = x + 8\).

Substitute \(-13\) for \(x\):

\(-5 = (-13) + 8\)

Now, simplify the right side:

\(-13 + 8 = -5\)

The left side is \(-5\) and the simplified right side is also \(-5\). Both sides are equal, confirming that our solution \(-13\) is correct.

This final step is crucial because it validates that all the manipulations we performed on the equation did not distort its truthfulness and our solution is indeed accurate.