Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To solve the problem, we need to determine how each of the given equations compares to the line [tex]\(a\)[/tex], which is represented by the equation [tex]\(y = -2x + 3\)[/tex].
### Step-by-Step Solution:
1. Identify the slope of line [tex]\(a\)[/tex]:
The slope-intercept form of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
For the line [tex]\(a\)[/tex], [tex]\(y = -2x + 3\)[/tex], the slope [tex]\(m\)[/tex] is [tex]\(-2\)[/tex].
2. Compare the slopes of the given equations to the slope of line [tex]\(a\)[/tex]:
- Equation 1: [tex]\(y = 2x - 1\)[/tex]
- The slope is [tex]\(2\)[/tex].
- Equation 2: [tex]\(y = -2x + 5\)[/tex]
- The slope is [tex]\(-2\)[/tex].
- Equation 3: [tex]\(y = \frac{1}{2}x + 7\)[/tex]
- The slope is [tex]\(\frac{1}{2}\)[/tex].
3. Determine the relationship of each equation to line [tex]\(a\)[/tex]:
- Parallel Lines:
- Two lines are parallel if they have the same slope.
- The slope of line [tex]\(a\)[/tex] is [tex]\(-2\)[/tex].
- The equation that has the same slope is [tex]\(y = -2x + 5\)[/tex].
- Therefore, [tex]\(y = -2x + 5\)[/tex] is parallel to line [tex]\(a\)[/tex].
- Perpendicular Lines:
- Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. This means their slopes are negative reciprocals of each other.
- The slope of line [tex]\(a\)[/tex] is [tex]\(-2\)[/tex].
- The negative reciprocal of [tex]\(-2\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
- The equation that has the slope [tex]\(\frac{1}{2}\)[/tex] is [tex]\(y = \frac{1}{2}x + 7\)[/tex].
- Therefore, [tex]\(y = \frac{1}{2}x + 7\)[/tex] is perpendicular to line [tex]\(a\)[/tex].
- Neither Parallel Nor Perpendicular:
- For an equation to be neither parallel nor perpendicular, its slope should not be the same as or the negative reciprocal of the slope of line [tex]\(a\)[/tex].
- The slope [tex]\(2\)[/tex] (from [tex]\(y = 2x - 1\)[/tex]) is neither [tex]\(-2\)[/tex] nor [tex]\(\frac{1}{2}\)[/tex].
- Therefore, [tex]\(y = 2x - 1\)[/tex] is neither parallel nor perpendicular to line [tex]\(a\)[/tex].
### Final Conclusion:
[tex]\[ \begin{tabular}{|c|c|c|} \hline \text{Parallel to line \(a\)} & \text{Perpendicular to line \(a\)} & \text{Neither parallel nor perpendicular to line \(a\)} \\ \hline y = -2x + 5 & y = \frac{1}{2}x + 7 & y = 2x - 1 \\ \hline \end{tabular} \][/tex]
### Step-by-Step Solution:
1. Identify the slope of line [tex]\(a\)[/tex]:
The slope-intercept form of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
For the line [tex]\(a\)[/tex], [tex]\(y = -2x + 3\)[/tex], the slope [tex]\(m\)[/tex] is [tex]\(-2\)[/tex].
2. Compare the slopes of the given equations to the slope of line [tex]\(a\)[/tex]:
- Equation 1: [tex]\(y = 2x - 1\)[/tex]
- The slope is [tex]\(2\)[/tex].
- Equation 2: [tex]\(y = -2x + 5\)[/tex]
- The slope is [tex]\(-2\)[/tex].
- Equation 3: [tex]\(y = \frac{1}{2}x + 7\)[/tex]
- The slope is [tex]\(\frac{1}{2}\)[/tex].
3. Determine the relationship of each equation to line [tex]\(a\)[/tex]:
- Parallel Lines:
- Two lines are parallel if they have the same slope.
- The slope of line [tex]\(a\)[/tex] is [tex]\(-2\)[/tex].
- The equation that has the same slope is [tex]\(y = -2x + 5\)[/tex].
- Therefore, [tex]\(y = -2x + 5\)[/tex] is parallel to line [tex]\(a\)[/tex].
- Perpendicular Lines:
- Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. This means their slopes are negative reciprocals of each other.
- The slope of line [tex]\(a\)[/tex] is [tex]\(-2\)[/tex].
- The negative reciprocal of [tex]\(-2\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
- The equation that has the slope [tex]\(\frac{1}{2}\)[/tex] is [tex]\(y = \frac{1}{2}x + 7\)[/tex].
- Therefore, [tex]\(y = \frac{1}{2}x + 7\)[/tex] is perpendicular to line [tex]\(a\)[/tex].
- Neither Parallel Nor Perpendicular:
- For an equation to be neither parallel nor perpendicular, its slope should not be the same as or the negative reciprocal of the slope of line [tex]\(a\)[/tex].
- The slope [tex]\(2\)[/tex] (from [tex]\(y = 2x - 1\)[/tex]) is neither [tex]\(-2\)[/tex] nor [tex]\(\frac{1}{2}\)[/tex].
- Therefore, [tex]\(y = 2x - 1\)[/tex] is neither parallel nor perpendicular to line [tex]\(a\)[/tex].
### Final Conclusion:
[tex]\[ \begin{tabular}{|c|c|c|} \hline \text{Parallel to line \(a\)} & \text{Perpendicular to line \(a\)} & \text{Neither parallel nor perpendicular to line \(a\)} \\ \hline y = -2x + 5 & y = \frac{1}{2}x + 7 & y = 2x - 1 \\ \hline \end{tabular} \][/tex]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
