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Isolating the variable is a crucial step in solving linear equations. It involves rearranging the equation so that the variable we are solving for stands alone on one side. This method helps in simplifying complex equations and identifying the variable's value.
In our example, we start with the equation: \ \( \frac{1}{2} x + 5 = -\frac{1}{2} x \).
To isolate the variable, we need to get all the \(x\) terms on one side. We do this by adding \( \frac{1}{2}x \) to both sides:
\ \( \frac{1}{2} x + 5 + \frac{1}{2} x = -\frac{1}{2} x + \frac{1}{2} x \).
This simplifies to: \ \( x + 5 = 0 \).
Now, the \( x \) term is isolated. This makes it easier to solve for \( x \), as we have reduced the equation to a simpler form.

Combining like terms is another essential concept in solving linear equations. It involves merging terms that have the same variable or reducing constant terms. This step simplifies the equation significantly.
In our current equation: \ \( x + 5 = 0 \), there are no more like terms to combine because we already have a single \( x \) term and a constant term.
Initially, we combined the \( \frac{1}{2}x \) terms on both sides of the original equation to yield one \( x \) term. This was necessary to reduce the equation to its simplest form.
For example, combining \( \frac{1}{2}x \) and \( \frac{1}{2}x \) to get \( x \). This helps streamline the process and solve the equation efficiently. Always look for like terms to combine them and simplify your work.

After solving any linear equation, it is crucial to check your solution. This confirms that your solution is correct and satisfies the original equation.
Let's verify our solution \( x = -5 \) by substituting it back into the original equation: \ \( \frac{1}{2}(-5) + 5 = -\frac{1}{2}(-5) \).
Simplifying the left side: \ \( -\frac{5}{2} + 5 \) converts to \( \frac{10}{2} - \frac{5}{2} = \frac{5}{2} \).
Simplifying the right side: \ \( \frac{5}{2} \).
Since both sides are equal (\( \frac{5}{2} = \frac{5}{2} \)), our solution \( x = -5 \) is verified as correct.
This step ensures accuracy and gives confidence that the solution truly addresses the original equation. Always recheck your solution to avoid any potential errors.