91Ó°ÊÓ

In algebra, isolating variables is one of the most important techniques. The goal is to get the variable (like x) by itself on one side of the equation. This makes it easier to solve for that variable. Let's look at an example where we have the equation: ewline $$ \frac{x}{5} = 15 $$ ewline In this equation, x is divided by 5. To isolate x, we need to eliminate the fraction. We can do this by multiplying both sides of the equation by 5. This 'undoes' the division by 5 and leaves us with the variable x on one side. ewline Always remember: whatever you do to one side of the equation, you must do to the other to keep the equation balanced. So, multiply both sides by 5: ewline ewline $$ \frac{x}{5} \times 5 = 15 \times 5 $$ ewline This simplifies to: ewline $$ x = 75 $$ ewline Now, the variable x is isolated, and we have its value.

Multiplication is a key operation when solving equations, especially for canceling out fractions. In our exercise, we had: ewline $$ \frac{x}{5} = 15 $$ ewline Here, x was divided by 5, and we needed to cancel out the fraction to isolate x. We did this by multiplying both sides of the equation by 5. ewline ewline $$ \frac{x}{5} \times 5 = 15 \times 5 $$ ewline Multiplying both sides by 5 helps to balance the equation while canceling the division by 5 on the left side. This step gives us a simpler equation: ewline $$ x = 75 $$ ewline By understanding this, you can apply multiplication to solve other similar equations easily. Remember, multiplication is just one way to manipulate equations to isolate variables and find solutions.

After isolating the variable and solving the equation, it's essential to check your solution to make sure it's correct. This ensures that no mistakes were made in the calculations. For our exercise, the solution we found was x = 75. To check this: ewline ewline Start by substituting x = 75 back into the original equation: ewline $$ \frac{x}{5} = 15 $$ ewline Substitute x: ewline $$ \frac{75}{5} = 15 $$ ewline ewline Simplify the left side: ewline $$ 15 = 15 $$ ewline Both sides of the equation match, verifying our solution is correct. ewline Checking solutions avoids errors and confirms that the solution satisfies the original equation. Always make it a habit to check your answers!