To determine the nature of the system of equations provided, we need to analyze their structure and solve them step by step.
Given the system of equations:
[tex]\[
\begin{array}{l}
3y = 9x - 6 \\
2y + 6x = 4
\end{array}
\][/tex]
First, let's rewrite both equations in the standard form [tex]\(Ax + By = C\)[/tex].
For the first equation:
[tex]\[
3y = 9x - 6 \implies 0x + 3y = 9x - 6 \implies -9x + 3y = -6 \implies -9x + 3y = -6 \implies 0x + 3y = 9
\][/tex]
So, the first equation can be written as:
[tex]\[
0x + 3y = 9
\][/tex]
For the second equation:
[tex]\[
2y + 6x = 4
\][/tex]
Let's make it look nicer:
[tex]\[
6x + 2y = 4
\][/tex]
Now we have the system in standard form:
[tex]\[
\begin{array}{l}
3y = 9, \ \ \text{or} \ 0x + 3y = 9\\
6x + 2y = 4
\end{array}
\][/tex]
Next, we need to find the determinant of the coefficient matrix to determine the type of system:
[tex]\[
\text{Determinant} = a_1 \cdot b_2 - a_2 \cdot b_1
\][/tex]
From the given system:
[tex]\[
a_1 = 0, b_1 = 3, c_1 = 9 \\
a_2 = 6, b_2 = 2, c_2 = 4
\][/tex]
Therefore,
[tex]\[
\text{Determinant} = 0 \cdot 2 - 6 \cdot 3 = 0 - 18 = -18
\][/tex]
Since the determinant is non-zero ([tex]\(-18\neq0\)[/tex]), this indicates that the system of equations is consistent and independent. This means that there is exactly one solution to this system where the lines intersect at a single point.
Hence, the pair of words that describes this system of equations is consistent and independent.
