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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]
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