To determine the nature of the given system of equations, let's analyze it step by step.
1. Given Equations:
[tex]\[
\begin{cases}
3y = 9x - 6 & \text{(1)} \\
2y + 6x = 4 & \text{(2)}
\end{cases}
\][/tex]
2. Rewriting Equation (1):
Let's rewrite Equation (1) in a more standard form:
[tex]\[
3y = 9x - 6
\][/tex]
Moving all terms involving variables to one side to get it into the standard linear form [tex]\(Ax + By = C\)[/tex], we get:
[tex]\[
9x - 3y = 6 \quad \text{(Equation 1')}
\][/tex]
3. Rewriting Equation (2):
Now, let's rewrite Equation (2) in standard form:
[tex]\[
2y + 6x = 4
\][/tex]
Rearranging to standard linear form [tex]\(Ax + By = C\)[/tex]:
[tex]\[
6x + 2y = 4 \quad \text{(Equation 2')}
\][/tex]
4. Matrix Form:
The two equations are:
[tex]\[
\begin{cases}
9x - 3y = 6 & \text{(1')} \\
6x + 2y = 4 & \text{(2')}
\end{cases}
\][/tex]
We now convert these into a matrix form [tex]\(A\mathbf{x} = \mathbf{b}\)[/tex]:
[tex]\[
\begin{pmatrix}
9 & -3 \\
6 & 2
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
6 \\
4
\end{pmatrix}
\][/tex]
5. Analyzing the System:
To analyze the system, we can check the determinant of the coefficient matrix. The determinant will indicate if the system has a unique solution, no solution, or infinite solutions.
Coefficient matrix:
[tex]\[
A = \begin{pmatrix}
9 & -3 \\
6 & 2
\end{pmatrix}
\][/tex]
The determinant of [tex]\(A\)[/tex] is calculated as:
[tex]\[
\det(A) = (9 \cdot 2) - (-3 \cdot 6) = 18 + 18 = 36
\][/tex]
Since the determinant is not zero ([tex]\(\det(A) \neq 0\)[/tex]), the coefficient matrix [tex]\(A\)[/tex] is invertible, indicating that the system of equations has a unique solution.
6. Conclusion:
Given that the determinant is non-zero, the system of equations has a unique solution. This means that the system is both consistent (it has at least one solution) and independent (it has exactly one unique solution).
Therefore, the pair of words that describe this system of equations is:
[tex]\[
\boxed{\text{consistent and independent}}
\][/tex]
