91Ó°ÊÓ

When solving equations, isolating the variable is the first and most important step. This means you want to get the variable by itself on one side of the equation. Let's see how it's done with our example: \( 10w - 5 = 65 \).
To isolate the variable term \(10w\), we first need to remove the constant term (which is -5) from the left side of the equation. We do this by adding 5 to both sides.

Here's the math behind it:
$$ 10w - 5 + 5 = 65 + 5 $$
We add the same number to both sides to keep the equation balanced.
This simplification gives us:
$$ 10w = 70 $$
Now, we have successfully isolated the variable term \(10w\). The next step is to solve for \(w\).

When you have variables on both sides of the equation, your first aim is to get all variables on one side. However, in our example, we don’t have variables on both sides since it’s already on one side. This simplifies our task.

In more complex problems where there are variables on both sides, you would:

• First, simplify each side if needed.
• Move all terms containing the variable to one side.
• Move all constants to the opposite side.

For instance, if we had an equation like:
\(10w - 5 = 2w + 13\),
We would start by subtracting \(2w\) from both sides:
$$ 10w - 2w - 5 = 2w - 2w + 13 $$
Simplifying to:
$$ 8w - 5 = 13 $$
Now, you can follow the same isolation steps as in the previous example.

When solving equations, following a step-by-step solution method helps you avoid mistakes and keeps your work organized. Let's review our earlier example with this method in mind.

Step 1: Isolate the variable term
In the equation \(10w - 5 = 65\), we added 5 to both sides to isolate the variable term. Simplifying gave us \(10w = 70\).
Step 2: Solve for the variable
Next, we divided both sides by 10 to solve for \(w\). Here's how it's done:
\(\frac{10w}{10} = \frac{70}{10}\)
This simplifies to:
\(w = 7\)

Recapping, a step-by-step approach includes:

• Performing the same operation on both sides.
• Keeping the equation balanced.
• Systematically simplifying the equation.
• Checking your solution by substituting back into the original equation.

Let’s apply this to ensure our answer is correct. Substituting \(w = 7\) back into the original equation:
\(10(7) - 5 = 65\), which simplifies to \(70 - 5 = 65\) confirms our solution is correct.