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To find the equation of the line that is perpendicular to the given line [tex]\(4x - 5y = 5\)[/tex] and passes through the point [tex]\((5,3)\)[/tex], follow these steps:
1. Convert the given line to its slope-intercept form to find its slope:
- The given line is [tex]\(4x - 5y = 5\)[/tex].
- Isolate [tex]\(y\)[/tex] on one side to convert it to the form [tex]\(y = mx + b\)[/tex].
- Start by subtracting [tex]\(4x\)[/tex] from both sides:
[tex]\[ -5y = -4x + 5 \][/tex]
- Divide each term by [tex]\(-5\)[/tex]:
[tex]\[ y = \frac{4}{5}x - 1 \][/tex]
- The slope [tex]\(m\)[/tex] of the given line is [tex]\(\frac{4}{5}\)[/tex].
2. Determine the slope of the perpendicular line:
- The slope of a line perpendicular to another is the negative reciprocal of the original slope.
- For the slope [tex]\(\frac{4}{5}\)[/tex], the negative reciprocal is [tex]\(-\frac{5}{4}\)[/tex].
3. Use the point-slope form of the equation of a line:
- The point-slope form is [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is the given point and [tex]\(m\)[/tex] is the slope of the line.
- Here, the point is [tex]\((5,3)\)[/tex] and the slope is [tex]\(-\frac{5}{4}\)[/tex].
4. Substitute the point and the slope into the point-slope form:
- [tex]\(y - 3 = -\frac{5}{4}(x - 5)\)[/tex].
5. Convert this equation to slope-intercept form (y = mx + b):
- Distribute the slope [tex]\(-\frac{5}{4}\)[/tex]:
[tex]\[ y - 3 = -\frac{5}{4}x + \frac{25}{4} \][/tex]
- Isolate [tex]\(y\)[/tex] by adding 3 to both sides:
[tex]\[ y = -\frac{5}{4}x + \frac{25}{4} + 3 \][/tex]
- Convert 3 to a fraction with the same denominator:
[tex]\[ y = -\frac{5}{4}x + \frac{25}{4} + \frac{12}{4} \][/tex]
- Combine the fractions:
[tex]\[ y = -\frac{5}{4}x + \frac{37}{4} \][/tex]
So, the equation of the line that is perpendicular to the given line and passes through the point [tex]\((5,3)\)[/tex] is [tex]\(y = -\frac{5}{4}x + \frac{37}{4}\)[/tex].
Given the multiple choices provided, the equivalent form of this equation is:
[tex]\[ 5x + 4y = 37 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{5 x + 4 y = 37} \][/tex]
1. Convert the given line to its slope-intercept form to find its slope:
- The given line is [tex]\(4x - 5y = 5\)[/tex].
- Isolate [tex]\(y\)[/tex] on one side to convert it to the form [tex]\(y = mx + b\)[/tex].
- Start by subtracting [tex]\(4x\)[/tex] from both sides:
[tex]\[ -5y = -4x + 5 \][/tex]
- Divide each term by [tex]\(-5\)[/tex]:
[tex]\[ y = \frac{4}{5}x - 1 \][/tex]
- The slope [tex]\(m\)[/tex] of the given line is [tex]\(\frac{4}{5}\)[/tex].
2. Determine the slope of the perpendicular line:
- The slope of a line perpendicular to another is the negative reciprocal of the original slope.
- For the slope [tex]\(\frac{4}{5}\)[/tex], the negative reciprocal is [tex]\(-\frac{5}{4}\)[/tex].
3. Use the point-slope form of the equation of a line:
- The point-slope form is [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is the given point and [tex]\(m\)[/tex] is the slope of the line.
- Here, the point is [tex]\((5,3)\)[/tex] and the slope is [tex]\(-\frac{5}{4}\)[/tex].
4. Substitute the point and the slope into the point-slope form:
- [tex]\(y - 3 = -\frac{5}{4}(x - 5)\)[/tex].
5. Convert this equation to slope-intercept form (y = mx + b):
- Distribute the slope [tex]\(-\frac{5}{4}\)[/tex]:
[tex]\[ y - 3 = -\frac{5}{4}x + \frac{25}{4} \][/tex]
- Isolate [tex]\(y\)[/tex] by adding 3 to both sides:
[tex]\[ y = -\frac{5}{4}x + \frac{25}{4} + 3 \][/tex]
- Convert 3 to a fraction with the same denominator:
[tex]\[ y = -\frac{5}{4}x + \frac{25}{4} + \frac{12}{4} \][/tex]
- Combine the fractions:
[tex]\[ y = -\frac{5}{4}x + \frac{37}{4} \][/tex]
So, the equation of the line that is perpendicular to the given line and passes through the point [tex]\((5,3)\)[/tex] is [tex]\(y = -\frac{5}{4}x + \frac{37}{4}\)[/tex].
Given the multiple choices provided, the equivalent form of this equation is:
[tex]\[ 5x + 4y = 37 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{5 x + 4 y = 37} \][/tex]
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