Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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.
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. 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 chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
