Holly finds that $(11 m-13 n+6 m n)-(10 m-7 n+3 m n)=m-20 n+9 m n$. What error did Holly make?
A. She only used the additive inverse of 10 m when combining like terms.
B. She added the polynomials instead of subtracting.
C. She only used the additive inverses of $-7 n$ and $3 m n$ when combining like terms.
D. She did not combine all like terms.
Answer (1)
Distribute the negative sign: ( 11 m − 13 n + 6 mn ) − ( 10 m − 7 n + 3 mn ) = 11 m − 13 n + 6 mn − 10 m + 7 n − 3 mn .
Combine like terms: 11 m − 10 m = m , − 13 n + 7 n = − 6 n , 6 mn − 3 mn = 3 mn .
The correct expression is m − 6 n + 3 mn .
Holly's error: She only used the additive inverses of − 7 n and 3 mn when combining like terms.
Explanation
Problem Analysis Let's analyze Holly's calculation and identify her mistake. The problem is to simplify the expression ( 11 m − 13 n + 6 mn ) − ( 10 m − 7 n + 3 mn ) .
Distributing the Negative Sign First, distribute the negative sign to each term inside the second parentheses: ( 11 m − 13 n + 6 mn ) − ( 10 m − 7 n + 3 mn ) = 11 m − 13 n + 6 mn − 10 m + 7 n − 3 mn
Combining Like Terms Next, combine the like terms:
For m terms: 11 m − 10 m = m
For n terms: − 13 n + 7 n = − 6 n
For mn terms: 6 mn − 3 mn = 3 mn So, the correct simplified expression is m − 6 n + 3 mn .
Comparing with Holly's Result Now, let's compare this correct expression with Holly's result, which is m − 20 n + 9 mn . We can see that Holly made errors in combining the n terms and the mn terms.
Identifying the Specific Errors Let's examine the n terms. Holly has − 20 n instead of − 6 n . This means she didn't correctly apply the additive inverse to − 7 n . She should have added 7 n to − 13 n , but she seems to have subtracted 7 n from − 13 n or made some other error in combining these terms.
Now let's examine the mn terms. Holly has 9 mn instead of 3 mn . This means she didn't correctly apply the additive inverse to 3 mn . She should have subtracted 3 mn from 6 mn , but she seems to have added them or made some other error in combining these terms.
Conclusion Based on this analysis, Holly only used the additive inverses of − 7 n and 3 mn when combining like terms.
Examples
When balancing a checkbook, it's crucial to correctly subtract expenses. If you incorrectly handle negative signs, like Holly did, you'll end up with the wrong balance. This principle applies to many real-life situations where accurate accounting of positive and negative values is essential, such as managing budgets, calculating profits and losses in business, or even understanding changes in temperature.
