To classify the system of equations, let's consider the given equations:
[tex]\[
\begin{array}{l}
1) -3x - y = -2 \\
2) 21x + 7y = 14
\end{array}
\][/tex]
First, let's simplify the second equation:
[tex]\[ 21x + 7y = 14 \][/tex]
We can divide everything by 7:
[tex]\[ 3x + y = 2 \][/tex]
Now we have the simplified form of the second equation:
[tex]\[ 3) 3x + y = 2 \][/tex]
Let's rewrite the first equation for clarity:
[tex]\[
1) -3x - y = -2
\][/tex]
We observe that the first equation [tex]\((-3x - y = -2)\)[/tex] can be rewritten as:
[tex]\[
-3x - y = -2
\][/tex]
Multiplying both sides of this equation by [tex]\(-1\)[/tex]:
[tex]\[
3x + y = 2
\][/tex]
Now compare it with the third equation:
[tex]\[ 3x + y = 2 \][/tex]
We find that both equations are identical. This means that the two equations are the same line expressed differently. Since they represent the same line, it indicates that there are infinitely many solutions, as both equations represent the same line in the coordinate system.
Thus, these lines are coincident.
So, the correct answer is:
coincident
