Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To determine which equations represent the line that is perpendicular to the line [tex]\(5x - 2y = -6\)[/tex] and passes through the point [tex]\((5, -4)\)[/tex], we will proceed with a step-by-step analysis.
### Finding the Perpendicular Slope
First, let's identify the slope of the given line [tex]\(5x - 2y = -6\)[/tex].
1. Rearrange the given equation into the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ 5x - 2y = -6 \implies -2y = -5x - 6 \implies y = \frac{5}{2}x + 3 \][/tex]
So, the slope [tex]\(m\)[/tex] of the given line is [tex]\(\frac{5}{2}\)[/tex].
2. The slope of a line perpendicular to this will be the negative reciprocal of [tex]\(\frac{5}{2}\)[/tex], which is:
[tex]\[ m_{\perpendicular} = -\frac{1}{m} = -\frac{1}{\frac{5}{2}} = -\frac{2}{5} \][/tex]
### Equations Passing Through [tex]\((5, -4)\)[/tex] and Perpendicular Line
Now, we use the point [tex]\((5, -4)\)[/tex] and the perpendicular slope [tex]\(-\frac{2}{5}\)[/tex] to form the potential equations.
#### Check each given option:
1. Option 1: [tex]\(y = -\frac{2}{5}x - 2\)[/tex]
- This equation indicates a slope of [tex]\(-\frac{2}{5}\)[/tex], which is correct.
- Substitute the point [tex]\((5, -4)\)[/tex] into [tex]\(y = -\frac{2}{5}x - 2\)[/tex]:
[tex]\[ -4 \stackrel{?}{=} -\frac{2}{5}(5) - 2 \Rightarrow -4 = -2 - 2 \Rightarrow -4 = -4 \][/tex]
This is true. Thus, this equation is a correct candidate.
2. Option 2: [tex]\(2x + 5y = -10\)[/tex]
- First, rearrange this into the slope-intercept form:
[tex]\[ 5y = -2x - 10 \implies y = -\frac{2}{5}x - 2 \][/tex]
- This represents a slope of [tex]\(-\frac{2}{5}\)[/tex] (neg. reciprocal of given slope).
- Verify with point [tex]\((5, -4)\)[/tex]:
[tex]\[ 2(5) + 5(-4) = 10 - 20 = -10 \][/tex]
This satisfies the equation. Hence, this is a correct candidate.
3. Option 3: [tex]\(2x - 5y = -10\)[/tex]
- Rearrange to slope-intercept form:
[tex]\[ -5y = -2x - 10 \implies y = \frac{2}{5}x + 2 \][/tex]
- This represents a slope of [tex]\(\frac{2}{5}\)[/tex] (same as given's neg. reciprocal).
- Check with point [tex]\((5, -4)\)[/tex]:
[tex]\[ 2(5) - 5(-4) = 10 + 20 = 30 \neq -10 \][/tex]
So, this equation does not satisfy the point criteria.
4. Option 4: [tex]\(y + 4 = -\frac{2}{5}(x - 5)\)[/tex]
- This is already in point-slope form:
[tex]\[ y + 4 = -\frac{2}{5}(x - 5) \][/tex]
- This form simply translates to the perpendicular slope and passes through [tex]\((5, -4)\)[/tex]:
[tex]\[ -4 + 4 = -\frac{2}{5}(5 - 5) \Rightarrow 0 = 0 \][/tex]
This satisfies, making it a correct candidate.
5. Option 5: [tex]\(y - 4 = \frac{5}{2}(x + 5)\)[/tex]
- This form gives a slope of [tex]\(\frac{5}{2}\)[/tex], which is not the perpendicular slope.
- It doesn't fit the criteria for perpendicular lines.
### Conclusion:
The following three equations represent the line that is perpendicular to [tex]\(5x - 2y = -6\)[/tex] and passes through [tex]\((5, -4)\)[/tex]:
- [tex]\(2x + 5y = -10\)[/tex]
- [tex]\(y + 4 = -\frac{2}{5}(x - 5)\)[/tex]
- [tex]\(y = -\frac{2}{5}x - 2\)[/tex]
### Finding the Perpendicular Slope
First, let's identify the slope of the given line [tex]\(5x - 2y = -6\)[/tex].
1. Rearrange the given equation into the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ 5x - 2y = -6 \implies -2y = -5x - 6 \implies y = \frac{5}{2}x + 3 \][/tex]
So, the slope [tex]\(m\)[/tex] of the given line is [tex]\(\frac{5}{2}\)[/tex].
2. The slope of a line perpendicular to this will be the negative reciprocal of [tex]\(\frac{5}{2}\)[/tex], which is:
[tex]\[ m_{\perpendicular} = -\frac{1}{m} = -\frac{1}{\frac{5}{2}} = -\frac{2}{5} \][/tex]
### Equations Passing Through [tex]\((5, -4)\)[/tex] and Perpendicular Line
Now, we use the point [tex]\((5, -4)\)[/tex] and the perpendicular slope [tex]\(-\frac{2}{5}\)[/tex] to form the potential equations.
#### Check each given option:
1. Option 1: [tex]\(y = -\frac{2}{5}x - 2\)[/tex]
- This equation indicates a slope of [tex]\(-\frac{2}{5}\)[/tex], which is correct.
- Substitute the point [tex]\((5, -4)\)[/tex] into [tex]\(y = -\frac{2}{5}x - 2\)[/tex]:
[tex]\[ -4 \stackrel{?}{=} -\frac{2}{5}(5) - 2 \Rightarrow -4 = -2 - 2 \Rightarrow -4 = -4 \][/tex]
This is true. Thus, this equation is a correct candidate.
2. Option 2: [tex]\(2x + 5y = -10\)[/tex]
- First, rearrange this into the slope-intercept form:
[tex]\[ 5y = -2x - 10 \implies y = -\frac{2}{5}x - 2 \][/tex]
- This represents a slope of [tex]\(-\frac{2}{5}\)[/tex] (neg. reciprocal of given slope).
- Verify with point [tex]\((5, -4)\)[/tex]:
[tex]\[ 2(5) + 5(-4) = 10 - 20 = -10 \][/tex]
This satisfies the equation. Hence, this is a correct candidate.
3. Option 3: [tex]\(2x - 5y = -10\)[/tex]
- Rearrange to slope-intercept form:
[tex]\[ -5y = -2x - 10 \implies y = \frac{2}{5}x + 2 \][/tex]
- This represents a slope of [tex]\(\frac{2}{5}\)[/tex] (same as given's neg. reciprocal).
- Check with point [tex]\((5, -4)\)[/tex]:
[tex]\[ 2(5) - 5(-4) = 10 + 20 = 30 \neq -10 \][/tex]
So, this equation does not satisfy the point criteria.
4. Option 4: [tex]\(y + 4 = -\frac{2}{5}(x - 5)\)[/tex]
- This is already in point-slope form:
[tex]\[ y + 4 = -\frac{2}{5}(x - 5) \][/tex]
- This form simply translates to the perpendicular slope and passes through [tex]\((5, -4)\)[/tex]:
[tex]\[ -4 + 4 = -\frac{2}{5}(5 - 5) \Rightarrow 0 = 0 \][/tex]
This satisfies, making it a correct candidate.
5. Option 5: [tex]\(y - 4 = \frac{5}{2}(x + 5)\)[/tex]
- This form gives a slope of [tex]\(\frac{5}{2}\)[/tex], which is not the perpendicular slope.
- It doesn't fit the criteria for perpendicular lines.
### Conclusion:
The following three equations represent the line that is perpendicular to [tex]\(5x - 2y = -6\)[/tex] and passes through [tex]\((5, -4)\)[/tex]:
- [tex]\(2x + 5y = -10\)[/tex]
- [tex]\(y + 4 = -\frac{2}{5}(x - 5)\)[/tex]
- [tex]\(y = -\frac{2}{5}x - 2\)[/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
