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Sagot :
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.
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.
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