To determine which pair of words describes the given system of equations:
[tex]\[
\begin{array}{l}
3y = 9x + 6 \\
2y - 6x = 4
\end{array}
\][/tex]
we need to follow these steps:
1. Rewrite the equations in standard form [tex]\(Ax + By = C\)[/tex]:
- For the first equation [tex]\(3y = 9x + 6\)[/tex]:
[tex]\[
\text{Divide both sides by 3: } y = 3x + 2
\][/tex]
Rearrange to standard form:
[tex]\[
-3x + y = 2
\][/tex]
- The second equation is already in standard form:
[tex]\[
2y - 6x = 4
\][/tex]
2. Express both equations together in standard form:
[tex]\[
-3x + y = 2
\][/tex]
[tex]\[
-6x + 2y = 4
\][/tex]
3. Compare the equations:
To make comparison easier, we can manipulate the first equation to have the same coefficients as the second.
Multiply the first equation by 2:
[tex]\[
-6x + 2y = 4
\][/tex]
Now, both equations look the same:
[tex]\[
-6x + 2y = 4
\][/tex]
[tex]\[
-6x + 2y = 4
\][/tex]
4. Identify relationship between the two equations:
Since both equations are identical, they represent the same line. This means:
- The system has infinitely many solutions.
- The two lines are not just parallel, they are coincident (they lie on top of each other).
Therefore, the system of equations is consistent (since there is at least one solution) and dependent (since the equations represent the same line).
Thus, the pair of words that best describes this system of equations is:
[tex]\[
\boxed{\text{consistent and dependent}}
\][/tex]
