- Subtracting a polynomial is equivalent to adding its additive inverse: \(A - B = A + (-B)\).
- Distribute the negative sign to each term of the polynomial \((6m - 7 + 2n)\), which results in \(-6m + 7 - 2n\).
- Rewrite the expression as the sum: \((4m - 5) + (-6m + 7 - 2n)\).
- Identify this expression as option a. The final answer is \(\boxed{a}\).
Explanation
- Analyzing the Problem and the Subtraction Property
We are tasked with finding the equivalent expression for the subtraction of polynomials given by \((4m - 5) - (6m - 7 + 2n)\). The problem involves understanding how to handle the subtraction of a polynomial expression. The core principle here is that subtracting an expression is equivalent to adding its additive inverse. This means that if we have an expression of the form \(A - B\), it can be rewritten as \(A + (-B)\). Here, \(A = (4m - 5)\) and \(B = (6m - 7 + 2n)\). To find the correct answer, we must distribute the negative sign across every term inside the parentheses of the expression being subtracted.
- Distributing the Negative Sign
To determine the additive inverse of the expression \(B = (6m - 7 + 2n)\), we distribute the negative sign to each term inside the parentheses. The negative sign effectively flips the sign of every term:
\(- (6m - 7 + 2n) = (-1) \cdot 6m - (-1) \cdot 7 + (-1) \cdot 2n\)
\(- (6m - 7 + 2n) = -6m - (-7) - 2n\)
\(- (6m - 7 + 2n) = -6m + 7 - 2n\)
Thus, the additive inverse of \(B\) is \((-6m + 7 - 2n)\).
- Final Conclusion and Choice Selection
By replacing the subtraction with the addition of the additive inverse, we can rewrite the original expression:
\((4m - 5) - (6m - 7 + 2n) = (4m - 5) + (-6m + 7 - 2n)\)
Now, we compare this result with the provided options:
A\((4m - 5) + (-6m + 7 - 2n)\)
B\((4m - 5) + (-6m - 7 - 2n)\)
C\((4m - 5) + (6m + 7 + 2n)\)
D\((4m - 5) + (-6m + 7 + 2n)\)
The expression matches option a. Therefore, the equivalent expression is \((4m - 5) + (-6m + 7 - 2n)\).
Examples
Imagine you are calculating your total expenses for a month. You have a base cost of \(\$(4m - 5)\) and you want to subtract a variable cost bundle of \(\$(6m - 7 + 2n)\) from it. Understanding how to correctly subtract these groups ensures you track your remaining budget accurately. In algebraic terms, this is identical to adding the negative of the variable costs to your base cost, which helps you manage your finances systematically and avoid simple errors in sign changes.