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To determine the equation of the line that passes through the point [tex]\((5, 3)\)[/tex] and is perpendicular to one of the given lines, we will follow these steps:
1. Calculate the slopes of the given lines:
- For the line [tex]\(4x - 5y = 5\)[/tex], rewrite it in slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ 4x - 5y = 5 \implies -5y = -4x + 5 \implies y = \frac{4}{5}x - 1 \][/tex]
Thus, the slope [tex]\(m_1 = \frac{4}{5}\)[/tex].
- For the line [tex]\(5x + 4y = 37\)[/tex], rewrite it:
[tex]\[ 5x + 4y = 37 \implies 4y = -5x + 37 \implies y = -\frac{5}{4}x + \frac{37}{4} \][/tex]
The slope [tex]\(m_2 = -\frac{5}{4}\)[/tex].
- For the line [tex]\(4x + 5y = 5\)[/tex], rewrite it:
[tex]\[ 4x + 5y = 5 \implies 5y = -4x + 5 \implies y = -\frac{4}{5}x + 1 \][/tex]
The slope [tex]\(m_3 = -\frac{4}{5}\)[/tex].
- For the line [tex]\(5x - 4y = 8\)[/tex], rewrite it:
[tex]\[ 5x - 4y = 8 \implies -4y = -5x + 8 \implies y = \frac{5}{4}x - 2 \][/tex]
The slope [tex]\(m_4 = \frac{5}{4}\)[/tex].
2. Determine the slopes of the perpendicular lines:
The slope of a line perpendicular to a given line with slope [tex]\(m\)[/tex] is [tex]\(-\frac{1}{m}\)[/tex].
- For [tex]\(m_1 = \frac{4}{5}\)[/tex], the perpendicular slope [tex]\(p_1 = -\frac{5}{4}\)[/tex].
- For [tex]\(m_2 = -\frac{5}{4}\)[/tex], the perpendicular slope [tex]\(p_2 = \frac{4}{5}\)[/tex].
- For [tex]\(m_3 = -\frac{4}{5}\)[/tex], the perpendicular slope [tex]\(p_3 = \frac{5}{4}\)[/tex].
- For [tex]\(m_4 = \frac{5}{4}\)[/tex], the perpendicular slope [tex]\(p_4 = -\frac{4}{5}\)[/tex].
3. Form the equations of the perpendicular lines using the point-slope form [tex]\(y - y_1 = m(x - x_1)\)[/tex]:
- For [tex]\(p_1 = -\frac{5}{4}\)[/tex] and point [tex]\((5, 3)\)[/tex]:
[tex]\[ y - 3 = -\frac{5}{4}(x - 5) \implies y - 3 = -\frac{5}{4}x + \frac{25}{4} \implies y = -\frac{5}{4}x + \frac{25}{4} + \frac{12}{4} \implies y = -\frac{5}{4}x + \frac{37}{4} \][/tex]
- For [tex]\(p_2 = \frac{4}{5}\)[/tex] and point [tex]\((5, 3)\)[/tex]:
[tex]\[ y - 3 = \frac{4}{5}(x - 5) \implies y - 3 = \frac{4}{5}x - 4 \implies y = \frac{4}{5}x - 1 \][/tex]
- For [tex]\(p_3 = \frac{5}{4}\)[/tex] and point [tex]\((5, 3)\)[/tex]:
[tex]\[ y - 3 = \frac{5}{4}(x - 5) \implies y - 3 = \frac{5}{4}x - \frac{25}{4} \implies y = \frac{5}{4}x - \frac{25}{4} + \frac{12}{4} \implies y = \frac{5}{4}x - 2 \][/tex]
- For [tex]\(p_4 = -\frac{4}{5}\)[/tex] and point [tex]\((5, 3)\)[/tex]:
[tex]\[ y - 3 = -\frac{4}{5}(x - 5) \implies y - 3 = -\frac{4}{5}x + 4 \implies y = -\frac{4}{5}x + 1 \][/tex]
4. Compare these equations with the given lines:
- The line with slope [tex]\(\frac{4}{5}\)[/tex] and passing through [tex]\((5,3)\)[/tex] is [tex]\(y = \frac{4}{5}x - 1\)[/tex], which matches the first given line [tex]\(4x - 5y = 5\)[/tex].
- The line with slope [tex]\(-\frac{5}{4}\)[/tex] and passing through [tex]\((5,3)\)[/tex] is [tex]\(y = -\frac{5}{4}x + \frac{37}{4}\)[/tex], which matches the second line [tex]\(5x + 4y = 37\)[/tex].
- The line with slope [tex]\(\frac{5}{4}\)[/tex] and passing through [tex]\((5,3)\)[/tex] is [tex]\(y = \frac{5}{4}x - 2\)[/tex], which matches the fourth line [tex]\(5x - 4y = 8\)[/tex].
- The line with slope [tex]\(-\frac{4}{5}\)[/tex] and passing through [tex]\((5,3)\)[/tex] is [tex]\(y = -\frac{4}{5}x + 1\)[/tex], which matches the third line [tex]\(4x + 5y = 5\)[/tex].
However, no perpendicular line perfectly parallels any of the given options through [tex]\((5,3)\)[/tex]. Thus, none of the conditions described match the requirement exactly. Therefore, the answer is None.
1. Calculate the slopes of the given lines:
- For the line [tex]\(4x - 5y = 5\)[/tex], rewrite it in slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ 4x - 5y = 5 \implies -5y = -4x + 5 \implies y = \frac{4}{5}x - 1 \][/tex]
Thus, the slope [tex]\(m_1 = \frac{4}{5}\)[/tex].
- For the line [tex]\(5x + 4y = 37\)[/tex], rewrite it:
[tex]\[ 5x + 4y = 37 \implies 4y = -5x + 37 \implies y = -\frac{5}{4}x + \frac{37}{4} \][/tex]
The slope [tex]\(m_2 = -\frac{5}{4}\)[/tex].
- For the line [tex]\(4x + 5y = 5\)[/tex], rewrite it:
[tex]\[ 4x + 5y = 5 \implies 5y = -4x + 5 \implies y = -\frac{4}{5}x + 1 \][/tex]
The slope [tex]\(m_3 = -\frac{4}{5}\)[/tex].
- For the line [tex]\(5x - 4y = 8\)[/tex], rewrite it:
[tex]\[ 5x - 4y = 8 \implies -4y = -5x + 8 \implies y = \frac{5}{4}x - 2 \][/tex]
The slope [tex]\(m_4 = \frac{5}{4}\)[/tex].
2. Determine the slopes of the perpendicular lines:
The slope of a line perpendicular to a given line with slope [tex]\(m\)[/tex] is [tex]\(-\frac{1}{m}\)[/tex].
- For [tex]\(m_1 = \frac{4}{5}\)[/tex], the perpendicular slope [tex]\(p_1 = -\frac{5}{4}\)[/tex].
- For [tex]\(m_2 = -\frac{5}{4}\)[/tex], the perpendicular slope [tex]\(p_2 = \frac{4}{5}\)[/tex].
- For [tex]\(m_3 = -\frac{4}{5}\)[/tex], the perpendicular slope [tex]\(p_3 = \frac{5}{4}\)[/tex].
- For [tex]\(m_4 = \frac{5}{4}\)[/tex], the perpendicular slope [tex]\(p_4 = -\frac{4}{5}\)[/tex].
3. Form the equations of the perpendicular lines using the point-slope form [tex]\(y - y_1 = m(x - x_1)\)[/tex]:
- For [tex]\(p_1 = -\frac{5}{4}\)[/tex] and point [tex]\((5, 3)\)[/tex]:
[tex]\[ y - 3 = -\frac{5}{4}(x - 5) \implies y - 3 = -\frac{5}{4}x + \frac{25}{4} \implies y = -\frac{5}{4}x + \frac{25}{4} + \frac{12}{4} \implies y = -\frac{5}{4}x + \frac{37}{4} \][/tex]
- For [tex]\(p_2 = \frac{4}{5}\)[/tex] and point [tex]\((5, 3)\)[/tex]:
[tex]\[ y - 3 = \frac{4}{5}(x - 5) \implies y - 3 = \frac{4}{5}x - 4 \implies y = \frac{4}{5}x - 1 \][/tex]
- For [tex]\(p_3 = \frac{5}{4}\)[/tex] and point [tex]\((5, 3)\)[/tex]:
[tex]\[ y - 3 = \frac{5}{4}(x - 5) \implies y - 3 = \frac{5}{4}x - \frac{25}{4} \implies y = \frac{5}{4}x - \frac{25}{4} + \frac{12}{4} \implies y = \frac{5}{4}x - 2 \][/tex]
- For [tex]\(p_4 = -\frac{4}{5}\)[/tex] and point [tex]\((5, 3)\)[/tex]:
[tex]\[ y - 3 = -\frac{4}{5}(x - 5) \implies y - 3 = -\frac{4}{5}x + 4 \implies y = -\frac{4}{5}x + 1 \][/tex]
4. Compare these equations with the given lines:
- The line with slope [tex]\(\frac{4}{5}\)[/tex] and passing through [tex]\((5,3)\)[/tex] is [tex]\(y = \frac{4}{5}x - 1\)[/tex], which matches the first given line [tex]\(4x - 5y = 5\)[/tex].
- The line with slope [tex]\(-\frac{5}{4}\)[/tex] and passing through [tex]\((5,3)\)[/tex] is [tex]\(y = -\frac{5}{4}x + \frac{37}{4}\)[/tex], which matches the second line [tex]\(5x + 4y = 37\)[/tex].
- The line with slope [tex]\(\frac{5}{4}\)[/tex] and passing through [tex]\((5,3)\)[/tex] is [tex]\(y = \frac{5}{4}x - 2\)[/tex], which matches the fourth line [tex]\(5x - 4y = 8\)[/tex].
- The line with slope [tex]\(-\frac{4}{5}\)[/tex] and passing through [tex]\((5,3)\)[/tex] is [tex]\(y = -\frac{4}{5}x + 1\)[/tex], which matches the third line [tex]\(4x + 5y = 5\)[/tex].
However, no perpendicular line perfectly parallels any of the given options through [tex]\((5,3)\)[/tex]. Thus, none of the conditions described match the requirement exactly. Therefore, the answer is None.
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