To determine the number of lines parallel to the line [tex]\(2x + 6y - 7 = 0\)[/tex] that have an intercept of 9 between the coordinate axes, let's proceed step by step.
1. Identify the Slope of the Given Line:
The general form of a line equation is [tex]\(Ax + By + C = 0\)[/tex]. We can convert this to the slope-intercept form [tex]\(y = mx + c\)[/tex] where [tex]\(m\)[/tex] is the slope and [tex]\(c\)[/tex] is the y-intercept.
Starting with the given line equation:
[tex]\[
2x + 6y - 7 = 0
\][/tex]
Rearrange it to solve for [tex]\(y\)[/tex]:
[tex]\[
6y = -2x + 7
\][/tex]
[tex]\[
y = -\frac{1}{3}x + \frac{7}{6}
\][/tex]
From this, we can see that the slope [tex]\(m\)[/tex] of the line is [tex]\(-\frac{1}{3}\)[/tex].
2. Expression of Parallel Lines:
Lines parallel to the given line will have the same slope, [tex]\(-\frac{1}{3}\)[/tex]. Therefore, any parallel line can be written in the form:
[tex]\[
y = -\frac{1}{3}x + c
\][/tex]
where [tex]\(c\)[/tex] is a constant that determines the line's vertical position.
3. Intercepts and their Sum:
The intercept between the coordinate axes means the sum of the x-intercept and y-intercept of the line is 9.
For the line [tex]\(y = -\frac{1}{3}x + c\)[/tex]:
- The x-intercept is found by setting [tex]\(y = 0\)[/tex]:
[tex]\[
0 = -\frac{1}{3}x + c \quad \Rightarrow \quad x = 3c
\][/tex]
- The y-intercept is simply [tex]\(c\)[/tex].
We need the sum of the x-intercept and y-intercept to be 9:
[tex]\[
x\text{-intercept} + y\text{-intercept} = 9
\][/tex]
[tex]\[
3c + c = 9
\][/tex]
[tex]\[
4c = 9
\][/tex]
[tex]\[
c = \frac{9}{4}
\][/tex]
4. Conclusion:
Since [tex]\(c = \frac{9}{4}\)[/tex] is a unique value, there is only one line that is parallel to the given line and satisfies the condition of having an intercept sum of 9 between the coordinate axes.
Thus, the number of such lines is:
[tex]\[
\boxed{1}
\][/tex]
Therefore, the answer is A) 1.
