91Ó°ÊÓ

The first crucial step in solving linear equations is to isolate the variable. This means getting the variable (usually represented as x) by itself on one side of the equation. Think of it as clearing everything around x, so its value stands alone.

In the given equation, 3x - 8 = 0, we want to isolate x. To do this, we perform an algebraic operation that helps us remove any constants or coefficients surrounding the variable. Here, we add 8 to both sides because the -8 is on the same side as our variable. Adding 8 cancels out the -8, simplifying our equation to 3x = 8.

Remember, we do the same operation on both sides to keep the equation balanced. Just like a balanced scale, whatever you do to one side, you must do to the other!

Algebraic manipulation involves changing the structure of your equations to make them easier to solve. This includes operations such as adding, subtracting, multiplying, or dividing terms in the equation. The goal is to move terms around to ultimately isolate the variable.

From our isolated variable equation, 3x = 8, we now need to solve for x. Since 3 is being multiplied by x, we perform the opposite operation, which is division. Thus, we divide both sides by 3:

\[\frac{3x}{3} = \frac{8}{3} \ x = \frac{8}{3}\]

This step-by-step manipulation ensures that x is completely isolated and we find its value.

The final concept involves verifying the solution. To do this, we simply plug our value of x back into the original equation and check if both sides of the equation are equal.

With our solution, x = \(\frac{8}{3}\), substitute it back into the original equation 3x - 8 = 0:

\[3 \times \frac{8}{3} - 8 = 0 \ 8 - 8 = 0\]

Since both sides are equal, our solution is verified as correct.

Now, let's compare it to the provided solution \(\frac{3}{8}\). If x were \(\frac{3}{8}\), substituting it into the equation would give:

\[3 \times \frac{3}{8} - 8 = 0 \ \frac{9}{8} - 8 eq 0\]

Notice, this does not satisfy the original equation, proving that the provided solution \(\frac{3}{8}\) is false.