To determine whether the given pair of lines is parallel, perpendicular, or neither, we start by analyzing the two given equations of the lines:
1. [tex]\(9x + 3y = 12\)[/tex]
2. [tex]\(6x + 2y = 9\)[/tex]
The first step is to rewrite these equations in their slope-intercept form ([tex]\(y = mx + b\)[/tex]) to identify the slopes of the lines. This form is helpful because, in slope-intercept form, the coefficient of [tex]\(x\)[/tex] represents the slope ([tex]\(m\)[/tex]) of the line.
### Step 1: Convert the first equation to slope-intercept form
Starting with:
[tex]\[9x + 3y = 12\][/tex]
We solve for [tex]\(y\)[/tex] by isolating [tex]\(y\)[/tex] on one side of the equation:
[tex]\[3y = -9x + 12\][/tex]
Now, divide both sides by 3 to get:
[tex]\[y = -3x + 4\][/tex]
Thus, the slope [tex]\(m_1\)[/tex] of the first line is [tex]\(-3\)[/tex].
### Step 2: Convert the second equation to slope-intercept form
Next, we take:
[tex]\[6x + 2y = 9\][/tex]
Similar to the previous step, solve for [tex]\(y\)[/tex]:
[tex]\[2y = -6x + 9\][/tex]
Divide both sides by 2:
[tex]\[y = -3x + \frac{9}{2}\][/tex]
Thus, the slope [tex]\(m_2\)[/tex] of the second line is also [tex]\(-3\)[/tex].
### Step 3: Compare the slopes of the lines
Now that we have the slopes, we can compare them:
- [tex]\(m_1 = -3\)[/tex]
- [tex]\(m_2 = -3\)[/tex]
### Step 4: Conclusion
If two lines have the same slope, they are parallel. Perpendicular lines would have slopes that are negative reciprocals of each other (e.g., [tex]\(m\)[/tex] and [tex]\(-\frac{1}{m}\)[/tex]).
Given that both slopes are equal, we conclude:
The lines are parallel.
Thus, the correct answer is:
A. parallel
