To determine the error Holly made, let us carefully examine the steps of combining like terms when subtracting the two polynomials.
We start with the given polynomials:
[tex]\[ P_1 = 11m - 13n + 6mn \][/tex]
[tex]\[ P_2 = 10m - 7n + 3mn \][/tex]
To find the result of the subtraction [tex]\( P_1 - P_2 \)[/tex], we subtract the coefficients of the corresponding like terms:
1. For the [tex]\(m\)[/tex] terms:
[tex]\[ 11m - 10m = 1m \][/tex]
2. For the [tex]\(n\)[/tex] terms:
[tex]\[ -13n - (-7n) = -13n + 7n = -6n \][/tex]
3. For the [tex]\(mn\)[/tex] terms:
[tex]\[ 6mn - 3mn = 3mn \][/tex]
Thus, the correct result of the subtraction is:
[tex]\[ 1m - 6n + 3mn \][/tex]
Given that the correct answer is [tex]\( m - 6n + 3mn \)[/tex] and Holly's answer is [tex]\( m - 20n + 9mn \)[/tex], we can identify the errors made by Holly:
1. Holly's result for the [tex]\( m \)[/tex] term is correct: [tex]\( 11m - 10m = 1m \)[/tex].
2. Holly's result for the [tex]\( n \)[/tex] term is incorrect: She should have [tex]\( -6n \)[/tex], but she got [tex]\( -20n \)[/tex].
3. Holly's result for the [tex]\( mn \)[/tex] term is also incorrect: She should have [tex]\( 3mn \)[/tex], but she got [tex]\( 9mn \)[/tex].
To understand where she went wrong:
1. Holly correctly used the additive inverse of [tex]\( 10m \)[/tex].
2. Holly incorrectly used the additive inverse when combining the [tex]\( n \)[/tex] terms. Instead of adding [tex]\( 7n \)[/tex] to [tex]\( -13n \)[/tex], which should give [tex]\( -6n \)[/tex], she must have subtracted it incorrectly to get [tex]\( -20n \)[/tex].
3. Holly incorrectly used the additive inverse when combining the [tex]\( mn \)[/tex] terms. Instead of subtracting [tex]\( 3mn \)[/tex] from [tex]\( 6mn \)[/tex], she may have incorrectly subtracted a different value or added incorrectly to get [tex]\( 9mn \)[/tex].
From these observations, the closest explanation for her error is:
She only used the additive inverses of [tex]\( -7n \)[/tex] and [tex]\( 3mn \)[/tex] when combining like terms.
In summary, Holly made errors by incorrectly subtracting the terms involving [tex]\( n \)[/tex] and [tex]\( mn \)[/tex], resulting in the wrong coefficients for those terms in her final polynomial.
