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Sagot :
To determine the relationship between the two given lines represented by the equations:
1. [tex]\( y = \frac{3}{4} x - 2 \)[/tex]
2. [tex]\( y + 2 = \frac{4}{3}(x - 2) \)[/tex]
we will follow these steps:
### Step 1: Rewrite the Second Equation in Slope-Intercept Form
First, let's rewrite the second equation in the standard slope-intercept form, [tex]\( y = mx + b \)[/tex].
Given:
[tex]\[ y + 2 = \frac{4}{3}(x - 2) \][/tex]
Distribute [tex]\(\frac{4}{3}\)[/tex]:
[tex]\[ y + 2 = \frac{4}{3}x - \frac{4}{3} \cdot 2 \][/tex]
[tex]\[ y + 2 = \frac{4}{3}x - \frac{8}{3} \][/tex]
Subtract 2 from both sides:
[tex]\[ y = \frac{4}{3}x - \frac{8}{3} - 2 \][/tex]
Subtract [tex]\(\frac{6}{3}\)[/tex] (which equals 2) from [tex]\(\frac{8}{3}\)[/tex]:
[tex]\[ y = \frac{4}{3}x - \frac{8}{3} - \frac{6}{3} \][/tex]
[tex]\[ y = \frac{4}{3}x - \frac{14}{3} \][/tex]
The second line in slope-intercept form is:
[tex]\[ y = \frac{4}{3}x - \frac{14}{3} \][/tex]
### Step 2: Identify the Slopes of Both Equations
Now compare the given equations in slope-intercept form:
1. [tex]\( y = \frac{3}{4} x - 2 \)[/tex]
2. [tex]\( y = \frac{4}{3} x - \frac{14}{3} \)[/tex]
From these equations, we see that:
- The slope of the first line ([tex]\(m_1\)[/tex]) is [tex]\(\frac{3}{4}\)[/tex]
- The slope of the second line ([tex]\(m_2\)[/tex]) is [tex]\(\frac{4}{3}\)[/tex]
### Step 3: Compare the Slopes
- Same Line or Parallel:
If [tex]\( m_1 = m_2 \)[/tex], the lines are either the same line (if the intercepts are equal) or parallel lines (if the intercepts are different).
Here, [tex]\(\frac{3}{4} \ne \frac{4}{3}\)[/tex], so they are not the same line or parallel.
- Perpendicular:
If [tex]\( m_1 \cdot m_2 = -1 \)[/tex], the lines are perpendicular.
[tex]\( \frac{3}{4} \cdot \frac{4}{3} = 1 \)[/tex], so the lines are not perpendicular.
- Intersecting but not Perpendicular:
If the lines are neither parallel nor perpendicular, they intersect at some point.
Since the slopes are different and their product is not -1, the lines [tex]\(\require{enclose} y = \frac{3}{4} x - 2 \)[/tex] and [tex]\( y = \frac{4}{3} x - \frac{14}{3} \)[/tex] intersect but are not perpendicular.
### Conclusion
Therefore, the two given lines represent:
(D) Intersecting, but not perpendicular
1. [tex]\( y = \frac{3}{4} x - 2 \)[/tex]
2. [tex]\( y + 2 = \frac{4}{3}(x - 2) \)[/tex]
we will follow these steps:
### Step 1: Rewrite the Second Equation in Slope-Intercept Form
First, let's rewrite the second equation in the standard slope-intercept form, [tex]\( y = mx + b \)[/tex].
Given:
[tex]\[ y + 2 = \frac{4}{3}(x - 2) \][/tex]
Distribute [tex]\(\frac{4}{3}\)[/tex]:
[tex]\[ y + 2 = \frac{4}{3}x - \frac{4}{3} \cdot 2 \][/tex]
[tex]\[ y + 2 = \frac{4}{3}x - \frac{8}{3} \][/tex]
Subtract 2 from both sides:
[tex]\[ y = \frac{4}{3}x - \frac{8}{3} - 2 \][/tex]
Subtract [tex]\(\frac{6}{3}\)[/tex] (which equals 2) from [tex]\(\frac{8}{3}\)[/tex]:
[tex]\[ y = \frac{4}{3}x - \frac{8}{3} - \frac{6}{3} \][/tex]
[tex]\[ y = \frac{4}{3}x - \frac{14}{3} \][/tex]
The second line in slope-intercept form is:
[tex]\[ y = \frac{4}{3}x - \frac{14}{3} \][/tex]
### Step 2: Identify the Slopes of Both Equations
Now compare the given equations in slope-intercept form:
1. [tex]\( y = \frac{3}{4} x - 2 \)[/tex]
2. [tex]\( y = \frac{4}{3} x - \frac{14}{3} \)[/tex]
From these equations, we see that:
- The slope of the first line ([tex]\(m_1\)[/tex]) is [tex]\(\frac{3}{4}\)[/tex]
- The slope of the second line ([tex]\(m_2\)[/tex]) is [tex]\(\frac{4}{3}\)[/tex]
### Step 3: Compare the Slopes
- Same Line or Parallel:
If [tex]\( m_1 = m_2 \)[/tex], the lines are either the same line (if the intercepts are equal) or parallel lines (if the intercepts are different).
Here, [tex]\(\frac{3}{4} \ne \frac{4}{3}\)[/tex], so they are not the same line or parallel.
- Perpendicular:
If [tex]\( m_1 \cdot m_2 = -1 \)[/tex], the lines are perpendicular.
[tex]\( \frac{3}{4} \cdot \frac{4}{3} = 1 \)[/tex], so the lines are not perpendicular.
- Intersecting but not Perpendicular:
If the lines are neither parallel nor perpendicular, they intersect at some point.
Since the slopes are different and their product is not -1, the lines [tex]\(\require{enclose} y = \frac{3}{4} x - 2 \)[/tex] and [tex]\( y = \frac{4}{3} x - \frac{14}{3} \)[/tex] intersect but are not perpendicular.
### Conclusion
Therefore, the two given lines represent:
(D) Intersecting, but not perpendicular
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