To determine the nature of the system of equations given by:
[tex]\[
\begin{array}{l}
3y = 9x - 6 \\
2y - 6x = 4
\end{array}
\][/tex]
we need to analyze these equations step-by-step.
### Step 1: Simplify and Write Equations in Standard Form
First, let's try to write both equations in a standard form [tex]\(ax + by = c\)[/tex].
#### Equation 1:
[tex]\[3y = 9x - 6\][/tex]
Rewriting it we get:
[tex]\[9x - 3y = 6\][/tex]
We will call this Equation (1).
#### Equation 2:
[tex]\[2y - 6x = 4\][/tex]
We will call this Equation (2).
### Step 2: Observe and Compare
To determine the type of system, we should look at the coefficients of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and the constants:
- For Equation (1): [tex]\(9x - 3y = 6\)[/tex]
- Coefficient of [tex]\(x\)[/tex]: 9
- Coefficient of [tex]\(y\)[/tex]: -3
- Constant term: 6
- For Equation (2): [tex]\(2y - 6x = 4\)[/tex]
- Coefficient of [tex]\(x\)[/tex]: -6
- Coefficient of [tex]\(y\)[/tex]: 2
- Constant term: 4
### Step 3: Analyze the Ratios
We check the ratios of the coefficients of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and constants from both equations:
- Ratio of [tex]\(x\)[/tex] coefficients: [tex]\(\frac{9}{-6} = -1.5\)[/tex]
- Ratio of [tex]\(y\)[/tex] coefficients: [tex]\(\frac{-3}{2} = -1.5\)[/tex]
- Ratio of constants: [tex]\(\frac{6}{4} = 1.5\)[/tex]
Since:
[tex]\[
\frac{9}{-6} = \frac{-3}{2} \neq \frac{6}{4}
\][/tex]
### Step 4: Conclusion
The ratios of the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are equal, while the ratio of the constants is different. This means that the two equations describe parallel lines, which means they never intersect.
### Result
Since the lines are parallel but not the same line, there are no points that satisfy both equations simultaneously. Hence, this system of equations is inconsistent.
Therefore, the pair of words that describe this system of equations is Inconsistent.
