To find the values of [tex]\( m \)[/tex] and [tex]\( n \)[/tex] given the vectors [tex]\( p \)[/tex] and [tex]\( q \)[/tex], let's work with the condition [tex]\( p = q \)[/tex].
The vectors are defined as:
[tex]\[ p = \begin{pmatrix} m + 3 \\ 2 - n \end{pmatrix} \][/tex]
[tex]\[ q = \begin{pmatrix} 3m - 1 \\ n - 8 \end{pmatrix} \][/tex]
Since [tex]\( p = q \)[/tex], the corresponding components of the two vectors must be equal. This gives us two equations:
[tex]\[ m + 3 = 3m - 1 \][/tex]
[tex]\[ 2 - n = n - 8 \][/tex]
### Step-by-Step Solution:
1. Solving the first equation:
[tex]\[ m + 3 = 3m - 1 \][/tex]
Subtract [tex]\( m \)[/tex] from both sides to isolate the terms involving [tex]\( m \)[/tex] on one side:
[tex]\[ 3 = 2m - 1 \][/tex]
Add 1 to both sides:
[tex]\[ 4 = 2m \][/tex]
Divide both sides by 2:
[tex]\[ m = 2 \][/tex]
2. Solving the second equation:
[tex]\[ 2 - n = n - 8 \][/tex]
Add [tex]\( n \)[/tex] to both sides to combine [tex]\( n \)[/tex] on one side:
[tex]\[ 2 = 2n - 8 \][/tex]
Add 8 to both sides:
[tex]\[ 10 = 2n \][/tex]
Divide both sides by 2:
[tex]\[ n = 5 \][/tex]
### Conclusion:
The values of [tex]\( m \)[/tex] and [tex]\( n \)[/tex] that satisfy the given condition [tex]\( p = q \)[/tex] are:
[tex]\[ m = 2 \][/tex]
[tex]\[ n = 5 \][/tex]
