Simplify the expression completely:
$12 x+5-7 y-x+2 y+4=$
The given expression is \(12x + 5 - 7y - x + 2y + 4\). Our goal is to simplify this expression by grouping and combining all like terms. Like terms are those that contain the same variables raised to the same exponents, along with constant terms that have no variables. The expression consists of terms involving \(x\), terms involving \(y\), and constant numbers.
First, we group the terms that contain the variable \(x\). These are \(12x\) and \(-x\). Adding these coefficients gives us \(12x - 1x = (12 - 1)x = 11x\).
Next, we group the terms that contain the variable \(y\). These are \(-7y\) and \(2y\). Adding these coefficients gives us \(-7y + 2y = (-7 + 2)y = -5y\).
Finally, we group the constant terms that do not have any variables attached. These are \(5\) and \(4\). Summing these constants gives us \(5 + 4 = 9\).
By combining the results from the previous steps, we get the simplified expression: \(11x - 5y + 9\). Thus, the fully simplified expression is \(\boxed{11x - 5y + 9}\).
In real-world inventory management, imagine you receive a shipment of \(12\) boxes of apples and then misplace \(1\) box, while simultaneously tracking two types of fruit crates (\(y\) being orange crates, \(x\) being apple crates). If you start with a specific number of items and organize them into types, you are essentially performing the same algebraic simplification. By combining like items—like \(x\) for apples and \(y\) for oranges—you can quickly determine your net inventory, just as we simplified the expression to understand the net count of variables \(x\) and \(y\).