To determine the value of [tex]\( p \)[/tex] for which the system of equations
[tex]\[ p x + 3 y = p - 3 \][/tex]
[tex]\[ 12 x + p y = p \][/tex]
will be inconsistent, we need to analyze the consistency conditions for linear systems. The system of equations will be inconsistent if the determinant of the coefficient matrix is zero because this indicates that the system has no unique solution.
First, we rewrite the system in matrix form as:
[tex]\[
\begin{pmatrix}
p & 3 \\
12 & p
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
p - 3 \\
p
\end{pmatrix}
\][/tex]
The determinant of the coefficient matrix (which is the matrix of the coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]) must be zero for the system to be inconsistent. The determinant of a [tex]\( 2 \times 2 \)[/tex] matrix [tex]\(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)[/tex] is given by:
[tex]\[
\text{Determinant} = ad - bc
\][/tex]
For the given matrix:
[tex]\[
\begin{pmatrix}
p & 3 \\
12 & p
\end{pmatrix}
\][/tex]
The determinant is:
[tex]\[
\text{Determinant} = (p \cdot p) - (12 \cdot 3)
\][/tex]
Simplifying the expression:
[tex]\[
\text{Determinant} = p^2 - 36
\][/tex]
We need this determinant to be zero for the system to be inconsistent:
[tex]\[
p^2 - 36 = 0
\][/tex]
Solving for [tex]\( p \)[/tex]:
[tex]\[
p^2 = 36
\][/tex]
Taking the square root of both sides:
[tex]\[
p = \pm 6
\][/tex]
So the values of [tex]\( p \)[/tex] that make the determinant zero are [tex]\( p = 6 \)[/tex] and [tex]\( p = -6 \)[/tex].
To summarize, the system of equations will be inconsistent for [tex]\( p = 6 \)[/tex] and [tex]\( p = -6 \)[/tex].
