Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

(a) Given the vectors [tex]p = \begin{pmatrix} m+3 \\ 2-n \end{pmatrix} \text{ and } q = \begin{pmatrix} 3m-1 \\ n-8 \end{pmatrix}[/tex] and that [tex]p = q[/tex], find the values of [tex]m[/tex] and [tex]n[/tex].

Sagot :

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.