Let's solve the system of linear equations:
[tex]\[
\begin{cases}
7p - q = 2 \\
-21p + 3q = 5
\end{cases}
\][/tex]
First, let's rewrite the equations for clarity:
1. [tex]\( 7p - q = 2 \)[/tex]
2. [tex]\( -21p + 3q = 5 \)[/tex]
Step 1: Eliminate one of the variables
We can start by eliminating [tex]\(q\)[/tex]. Let's manipulate the first equation so that the coefficients of [tex]\(q\)[/tex] match in both equations.
Multiply the first equation by 3:
[tex]\[
3(7p - q) = 3 \cdot 2
\][/tex]
This gives us:
[tex]\[
21p - 3q = 6
\][/tex]
Now the system is:
[tex]\[
\begin{cases}
21p - 3q = 6 \\
-21p + 3q = 5
\end{cases}
\][/tex]
Step 2: Add the equations
[tex]\[
(21p - 3q) + (-21p + 3q) = 6 + 5
\][/tex]
Simplifying this:
[tex]\[
0 = 11
\][/tex]
This equation is a contradiction, which means that there is no solution to this system of equations. The equations are inconsistent.
Therefore, the system of equations:
[tex]\[
\begin{cases}
7p - q = 2 \\
-21p + 3q = 5
\end{cases}
\][/tex]
has no solution.
