IDNLearn.com provides a collaborative environment for finding and sharing knowledge. Get the information you need from our community of experts who provide accurate and comprehensive answers to all your questions.
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
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.
