Which value of $n$ makes the equation true?
$-\frac{1}{2} n=-8$
The problem asks us to solve the linear equation \(-\frac{1}{2} n = -8\) for the variable \(n\). We have four potential options to choose from: \(-16, -4, 4, 16\). Our goal is to isolate \(n\) to find its value and determine which of the provided choices is correct.
To solve for \(n\), we need to isolate it on one side of the equation. Currently, \(n\) is being multiplied by \(-\frac{1}{2}\). To undo this multiplication, we multiply both sides of the equation by the reciprocal of \(-\frac{1}{2}\), which is \(-2\). This gives us:
\[-\frac{1}{2} n \times (-2) = -8 \times (-2)\]
Calculating the left side, \((-\frac{1}{2}) \times (-2) = 1\), so we are left with \(n\) on the left side of the equation.
Calculating the right side, \((-8) \times (-2) = 16\).
Therefore, \(n = 16\).
Now, we substitute \(n = 16\) back into the original equation to verify that it satisfies the condition:
\[-\frac{1}{2} (16) = -8\]
Since \(-\frac{1}{2} \times 16 = -8\), the equality holds true. Comparing this result with the given options \(-16, -4, 4, 16\), we see that \(16\) is the correct choice.
In everyday life, linear equations are useful for determining unknown quantities. For instance, if you are splitting a total cost or adjusting a quantity based on a ratio, such as when a recipe calls for half the amount of an ingredient and you have a total of 8 servings remaining, you can use these principles to find the original amount. Here, the relationship between the scaling factor and the total amount is exactly what we solved.