Answer :
Alright class, let's take a detailed look at the given equations and determine the relationship between the two lines.
We start with the equations:
1. [tex]\(3x + 12y = 9\)[/tex]
2. [tex]\(2x - 8y = 4\)[/tex]
To understand the relationship, we'll need to find the slopes of these lines. We'll convert each equation to slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope.
### Step-by-Step Solution:
#### For the first equation: [tex]\(3x + 12y = 9\)[/tex]
1. Isolate [tex]\(y\)[/tex]:
[tex]\[ 12y = -3x + 9 \][/tex]
2. Divide every term by 12:
[tex]\[ y = -\frac{3}{12}x + \frac{9}{12} \][/tex]
3. Simplify the fractions:
[tex]\[ y = -\frac{1}{4}x + \frac{3}{4} \][/tex]
So, the slope [tex]\(m_1\)[/tex] of the first line is [tex]\(-\frac{1}{4}\)[/tex].
#### For the second equation: [tex]\(2x - 8y = 4\)[/tex]
1. Isolate [tex]\(y\)[/tex]:
[tex]\[ -8y = -2x + 4 \][/tex]
2. Divide every term by [tex]\(-8\)[/tex]:
[tex]\[ y = \frac{-2}{-8}x + \frac{4}{-8} \][/tex]
3. Simplify the fractions:
[tex]\[ y = \frac{1}{4}x - \frac{1}{2} \][/tex]
So, the slope [tex]\(m_2\)[/tex] of the second line is [tex]\(\frac{1}{4}\)[/tex].
### Analyzing the Slopes
Now that we have the slopes:
- [tex]\(m_1 = -\frac{1}{4}\)[/tex]
- [tex]\(m_2 = \frac{1}{4}\)[/tex]
Let's check if the lines are either parallel or perpendicular:
1. Parallel Lines: Two lines are parallel if their slopes are equal. Here, [tex]\(-\frac{1}{4}\)[/tex] is not equal to [tex]\(\frac{1}{4}\)[/tex], so the lines are not parallel.
2. Perpendicular Lines: Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. Let's calculate the product of the slopes:
[tex]\[ \left( -\frac{1}{4} \right) \times \left( \frac{1}{4} \right) = -\frac{1}{16} \][/tex]
Since [tex]\(-\frac{1}{16}\)[/tex] is not equal to [tex]\(-1\)[/tex], the lines are also not perpendicular.
### Conclusion
Since the slopes of the lines are not equal, the lines are not parallel. Furthermore, the product of their slopes is [tex]\(-\frac{1}{16}\)[/tex] rather than [tex]\(-1\)[/tex], so they are not perpendicular either.
Hence, the lines are neither parallel nor perpendicular.
We start with the equations:
1. [tex]\(3x + 12y = 9\)[/tex]
2. [tex]\(2x - 8y = 4\)[/tex]
To understand the relationship, we'll need to find the slopes of these lines. We'll convert each equation to slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope.
### Step-by-Step Solution:
#### For the first equation: [tex]\(3x + 12y = 9\)[/tex]
1. Isolate [tex]\(y\)[/tex]:
[tex]\[ 12y = -3x + 9 \][/tex]
2. Divide every term by 12:
[tex]\[ y = -\frac{3}{12}x + \frac{9}{12} \][/tex]
3. Simplify the fractions:
[tex]\[ y = -\frac{1}{4}x + \frac{3}{4} \][/tex]
So, the slope [tex]\(m_1\)[/tex] of the first line is [tex]\(-\frac{1}{4}\)[/tex].
#### For the second equation: [tex]\(2x - 8y = 4\)[/tex]
1. Isolate [tex]\(y\)[/tex]:
[tex]\[ -8y = -2x + 4 \][/tex]
2. Divide every term by [tex]\(-8\)[/tex]:
[tex]\[ y = \frac{-2}{-8}x + \frac{4}{-8} \][/tex]
3. Simplify the fractions:
[tex]\[ y = \frac{1}{4}x - \frac{1}{2} \][/tex]
So, the slope [tex]\(m_2\)[/tex] of the second line is [tex]\(\frac{1}{4}\)[/tex].
### Analyzing the Slopes
Now that we have the slopes:
- [tex]\(m_1 = -\frac{1}{4}\)[/tex]
- [tex]\(m_2 = \frac{1}{4}\)[/tex]
Let's check if the lines are either parallel or perpendicular:
1. Parallel Lines: Two lines are parallel if their slopes are equal. Here, [tex]\(-\frac{1}{4}\)[/tex] is not equal to [tex]\(\frac{1}{4}\)[/tex], so the lines are not parallel.
2. Perpendicular Lines: Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. Let's calculate the product of the slopes:
[tex]\[ \left( -\frac{1}{4} \right) \times \left( \frac{1}{4} \right) = -\frac{1}{16} \][/tex]
Since [tex]\(-\frac{1}{16}\)[/tex] is not equal to [tex]\(-1\)[/tex], the lines are also not perpendicular.
### Conclusion
Since the slopes of the lines are not equal, the lines are not parallel. Furthermore, the product of their slopes is [tex]\(-\frac{1}{16}\)[/tex] rather than [tex]\(-1\)[/tex], so they are not perpendicular either.
Hence, the lines are neither parallel nor perpendicular.
