Sure, let's solve the system of equations step by step given that the vectors [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are defined as follows and [tex]\( p = q \)[/tex]:
[tex]\[ p = \begin{pmatrix} m + 3 \\ 2 - n \end{pmatrix}, \quad q = \begin{pmatrix} 3m - 1 \\ n - 8 \end{pmatrix} \][/tex]
Since [tex]\( p = q \)[/tex], we equate the corresponding components of the vectors:
[tex]\[ \begin{pmatrix} m + 3 \\ 2 - n \end{pmatrix} = \begin{pmatrix} 3m - 1 \\ n - 8 \end{pmatrix} \][/tex]
This gives us two equations:
1. [tex]\( m + 3 = 3m - 1 \)[/tex]
2. [tex]\( 2 - n = n - 8 \)[/tex]
Now, solve each equation one by one.
Equation 1:
[tex]\[ m + 3 = 3m - 1 \][/tex]
Subtract [tex]\( m \)[/tex] from both sides to begin simplifying:
[tex]\[ 3 = 2m - 1 \][/tex]
Next, add 1 to both sides:
[tex]\[ 4 = 2m \][/tex]
Divide both sides by 2:
[tex]\[ m = 2 \][/tex]
Equation 2:
[tex]\[ 2 - n = n - 8 \][/tex]
Add [tex]\( n \)[/tex] to both sides to combine like terms:
[tex]\[ 2 = 2n - 8 \][/tex]
Next, add 8 to both sides:
[tex]\[ 10 = 2n \][/tex]
Divide both sides by 2:
[tex]\[ n = 5 \][/tex]
Thus, the values of [tex]\( m \)[/tex] and [tex]\( n \)[/tex] that satisfy the equations are:
[tex]\[ m = 2 \quad \text{and} \quad n = 5 \][/tex]
Feel free to verify these values by substituting [tex]\( m \)[/tex] and [tex]\( n \)[/tex] back into the original vectors to check if [tex]\( p = q \)[/tex]:
For [tex]\( m = 2 \)[/tex] and [tex]\( n = 5 \)[/tex]:
[tex]\[ p = \begin{pmatrix} 2 + 3 \\ 2 - 5 \end{pmatrix} = \begin{pmatrix} 5 \\ -3 \end{pmatrix} \][/tex]
[tex]\[ q = \begin{pmatrix} 3(2) - 1 \\ 5 - 8 \end{pmatrix} = \begin{pmatrix} 6 - 1 \\ -3 \end{pmatrix} = \begin{pmatrix} 5 \\ -3 \end{pmatrix} \][/tex]
Indeed, [tex]\( p = q \)[/tex], confirming that the solutions [tex]\( m = 2 \)[/tex] and [tex]\( n = 5 \)[/tex] are correct.
