What is the value of $g$?
$-5+7g = 3g+9$
The problem asks to solve the linear equation \(-5 + 7g = 3g + 9\) for the variable \(g\). A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable raised to the first power. To solve for \(g\), we need to isolate it on one side of the equation.
To isolate \(g\), we first gather all terms containing \(g\) on the left side of the equation. We do this by subtracting \(3g\) from both sides:
\[-5 + 7g - 3g = 3g - 3g + 9\]
\[-5 + 4g = 9\]
Next, we isolate the term with \(g\) by moving the constant term \(-5\) to the right side of the equation. We do this by adding \(5\) to both sides:
\[-5 + 5 + 4g = 9 + 5\]
\[4g = 14\]
Finally, we solve for \(g\) by dividing both sides of the equation by \(4\):
\[g = \frac{14}{4}\]
Simplifying the fraction by dividing the numerator and denominator by their greatest common divisor, which is \(2\), we get:
\[g = \frac{7}{2}\]
Comparing this result to the given options, we find that the value \(\frac{7}{2}\) matches the fourth option.
Imagine you are managing a small business where the revenue is calculated by a linear equation based on the number of units sold. If your cost structure is represented by \(-5 + 7g = 3g + 9\), where \(g\) represents the number of units to reach the break-even point, solving for \(g\) allows you to determine exactly how many units you need to sell to balance your budget. This simple algebraic process is foundational for financial planning, inventory management, and operational efficiency in various professional settings.