To solve the system of linear equations:
[tex]\[
\begin{cases}
7p - q = 2 \\
-21p + 3q = 5
\end{cases}
\][/tex]
we can use the method of substitution or the method of elimination. Here, let's use the method of elimination to find the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex].
### Step 1: Align the system of equations:
[tex]\[
\begin{array}{c}
7p - q = 2 \\
-21p + 3q = 5
\end{array}
\][/tex]
### Step 2: Let's multiply the first equation by a constant such that the coefficient of [tex]\( p \)[/tex] in both equations is the same. We choose to multiply the first equation by 3:
[tex]\[
3 \cdot (7p - q) = 3 \cdot 2 \\
21p - 3q = 6
\][/tex]
So the system now looks like:
[tex]\[
\begin{cases}
21p - 3q = 6 \\
-21p + 3q = 5
\end{cases}
\][/tex]
### Step 3: Add the two equations to eliminate [tex]\( q \)[/tex]:
[tex]\[
(21p - 3q) + (-21p + 3q) = 6 + 5 \\
0 = 11
\][/tex]
### Step 4: Since adding these equations gave us the result [tex]\( 0 = 11 \)[/tex], which is a contradiction, this means that there is no solution to the system of equations. The system is inconsistent.
Therefore, there are no values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] that satisfy both equations simultaneously.
