91Ó°ÊÓ

When solving equations, isolating the variable is a key step to finding its value. The variable we often talk about is usually represented by letters like \( x \). Isolating the variable means we want it to be by itself on one side of the equation. To do this, we need to work step-by-step to remove anything else that is attached to it.
Let's consider the equation from the exercise: \[-10x - 16 = 24\]You can see that the \( x \)-term is affected by other numbers. To isolate \( x \), we first eliminate the constant \(-16\) by adding \( 16 \) to both sides. This cancels out \(-16\) on the left side, like this:

• Add 16 to both sides: \(-10x - 16 + 16 = 24 + 16\)
• The equation simplifies: \(-10x = 40\)

Remember: Whatever you do to one side of the equation, you have to do to the other side as well. This is critical to maintaining the equation's balance.

Verifying the solution is an often overlooked but extremely important step in solving equations. This step ensures that the value we found for \( x \) is actually correct when substituted back into the original equation. Let's go through this final step using our solution of \( x = -4 \).First, we substitute \( x = -4 \) back into the original equation:\[-10(-4) - 16 = 24\]Next, solve the left side to verify that it equals the right side:

• Calculate \(-10\) times \(-4\) to get \( 40 \).
• Then, solve \( 40 - 16 = 24 \).
• This shows \( 24 = 24 \), confirming the solution.

This step proves beyond doubt that \( x = -4 \) is indeed the correct answer. Always verify your solutions to avoid mistakes and ensure mathematical accuracy.

To solve for \( x \), you perform operations that transform the equation step-by-step until \( x \) stands alone. In the given example, we'll start once you've isolated \( x \) as:\[-10x = 40\]Now focus on just the term with \( x \) and its coefficient, which is \(-10\). The goal is to get \( x \) by itself, so the next operation should neutralize this coefficient.Here's how you do it:

• Divide both sides by \(-10\): \( x = \frac{40}{-10} \)
• This simplifies to \( x = -4 \)

Dividing both sides by \(-10\) perfectly isolates \( x \) because

• It cancels out the multiplication by \(-10\)
• Leaves \( x \) as the sole term on one side.

Keep this in mind: Operations like addition, subtraction, multiplication, and division are your tools for unraveling the equation and zeroing in on \( x \).