91Ó°ÊÓ

Isolating the variable is an essential first step when solving an equation. It means getting the variable, like \( w \), by itself on one side of the equation. This allows you to solve for the unknown.

In our exercise, the initial equation is: \[ 0.4w + 12.8 = 16.2 \]
To isolate \( w \), we start by removing the constant term on the left-hand side (12.8) by subtracting it from both sides of the equation. This ensures the equation remains balanced.

So, subtract 12.8 from both sides:
\[ 0.4w + 12.8 - 12.8 = 16.2 - 12.8 \]
Simplifying this gives:
\[ 0.4w = 3.4 \]
Now the variable \( w \) is isolated, having been removed from additional numbers or constants on its side, making it easier to solve.

After finding the value of the variable, it's important to check your solution to ensure it is correct. Substitution is typically used for this step.

First, we need to substitute our solution back into the original equation. Our solution for the exercise was \( w = 8.5 \). Let's substitute this into the original equation: \[ 0.4(8.5) + 12.8 = 16.2 \]
Perform the multiplication: \[ 3.4 + 12.8 = 16.2 \]
Then, add the results on the left-hand side: \[ 16.2 = 16.2 \]
Since both sides of the equation are equal, this confirms that our solution is correct. Checking solutions is crucial because it helps verify that no mistakes were made during the process.

Simplifying an equation often means combining like terms and performing arithmetic operations to make the equation easier to solve. It also involves ensuring that the variables are isolated and constants are on one side.

In our exercise, after isolating \( w \): \[ 0.4w = 3.4 \]
We need to solve for \( w \). To do this, we divide both sides of the equation by the coefficient of \( w \) (0.4):
\[ \frac{0.4w}{0.4} = \frac{3.4}{0.4} \]
This simplifies the equation to:
\[ w = 8.5 \]
By performing these operations, we reduce the equation to its simplest form, making it clear and easy to understand. Simplification helps in quickly identifying the value of the unknown variable.