91Ó°ÊÓ

When solving linear equations, the goal is to find the value of the variable that makes the equation true. To do this efficiently, isolating the variable is crucial. Isolation means you want the variable on one side of the equation by itself.

Let's consider the equation \(-4x = -64\). Our variable here is either side of the equation. In our example, let's isolate x.

To isolate x, we need to get rid of the coefficient (the number multiplying the variable). This often involves using opposite operations, like addition, subtraction, multiplication, or division. Here, dividing both sides by -4 (the coefficient of x) will effectively isolate our variable.

After division, we get \( x = 16 \), successfully isolating the variable.

After finding the value of the variable, it's always an excellent idea to check your solution. This ensures you didn't make any mistakes during calculations.

How do you check a solution? Substitute the value of the variable back into the original equation and verify both sides are equal.

In our exercise, we found \( x = 16 \). Substituting 16 back into our original equation \(-4x = -64 \): \(-4(16)\).

Calculating -4 multiplied by 16 gives us -64, exactly the right-hand side of our original equation. Since the left side equals the right side, our solution is correct!

This is a straightforward check and helps ensure our solution is accurate before moving on to the next problem.

Division is a key operation in algebra, particularly when isolating a variable. Algebraic division follows similar rules as arithmetic but involves variables and coefficients.

Let's revisit our solved example: \(-4x = -64\). To solve for x, we divided both sides by -4: \(\frac{-64}{-4} \).

Division helps isolate the variable by removing the coefficient.

It's essential to remember:

• Always do the same operation on both sides to maintain equality.
• Opposite signs change (negative becomes positive and vice versa when dividing two negative numbers).
• If the coefficient is +1 or -1, dividing by 1 doesn't change the variable's value but still isolates it.

Division in algebra simplifies equations and is a practical tool for getting variables alone.