Answer :
To solve the problem of finding the equation of a line that is perpendicular to the given line [tex]\(3x - 5y = 20\)[/tex] and passes through the point [tex]\((-6, 4)\)[/tex], let's follow step-by-step instructions:
1. Convert the given line equation to slope-intercept form:
The given line equation is:
[tex]\[ 3x - 5y = 20 \][/tex]
To convert this to slope-intercept form ([tex]\(y = mx + b\)[/tex]), solve for [tex]\(y\)[/tex]:
[tex]\[ -5y = -3x + 20 \][/tex]
Divide by [tex]\(-5\)[/tex]:
[tex]\[ y = \frac{3}{5}x - 4 \][/tex]
Therefore, the corresponding slope [tex]\(m\)[/tex] of the given line is:
[tex]\[ m = \frac{3}{5} \][/tex]
2. Find the slope of the line perpendicular to the given line:
Perpendicular slopes are negative reciprocals of each other; therefore, if the slope of the given line is [tex]\(\frac{3}{5}\)[/tex], the slope [tex]\(m'\)[/tex] of the perpendicular line will be:
[tex]\[ m' = -\frac{1}{m} = -\frac{1}{\frac{3}{5}} = -\frac{5}{3} \][/tex]
3. Use the point-slope form of the equation of a line that passes through the point [tex]\((-6, 4)\)[/tex]:
The point-slope form of a line is given by:
[tex]\[ y - y_1 = m'(x - x_1) \][/tex]
Where [tex]\((x_1, y_1)\)[/tex] is [tex]\((-6, 4)\)[/tex], and [tex]\(m' = -\frac{5}{3}\)[/tex]:
[tex]\[ y - 4 = -\frac{5}{3} (x + 6) \][/tex]
4. Simplify the equation to slope-intercept form (if necessary):
Distribute [tex]\(-\frac{5}{3}\)[/tex]:
[tex]\[ y - 4 = -\frac{5}{3} x - 10 \][/tex]
Add [tex]\(4\)[/tex] to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[ y = -\frac{5}{3}x - 6 \][/tex]
5. Choose the correct answer:
From the given options:
[tex]\[ y = -\frac{5}{3}x - 6 \][/tex]
Matches exactly with the equation we derived. Thus, the correct answer is:
[tex]\[ y = -\frac{5}{3}x - 6 \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{3} \][/tex]
1. Convert the given line equation to slope-intercept form:
The given line equation is:
[tex]\[ 3x - 5y = 20 \][/tex]
To convert this to slope-intercept form ([tex]\(y = mx + b\)[/tex]), solve for [tex]\(y\)[/tex]:
[tex]\[ -5y = -3x + 20 \][/tex]
Divide by [tex]\(-5\)[/tex]:
[tex]\[ y = \frac{3}{5}x - 4 \][/tex]
Therefore, the corresponding slope [tex]\(m\)[/tex] of the given line is:
[tex]\[ m = \frac{3}{5} \][/tex]
2. Find the slope of the line perpendicular to the given line:
Perpendicular slopes are negative reciprocals of each other; therefore, if the slope of the given line is [tex]\(\frac{3}{5}\)[/tex], the slope [tex]\(m'\)[/tex] of the perpendicular line will be:
[tex]\[ m' = -\frac{1}{m} = -\frac{1}{\frac{3}{5}} = -\frac{5}{3} \][/tex]
3. Use the point-slope form of the equation of a line that passes through the point [tex]\((-6, 4)\)[/tex]:
The point-slope form of a line is given by:
[tex]\[ y - y_1 = m'(x - x_1) \][/tex]
Where [tex]\((x_1, y_1)\)[/tex] is [tex]\((-6, 4)\)[/tex], and [tex]\(m' = -\frac{5}{3}\)[/tex]:
[tex]\[ y - 4 = -\frac{5}{3} (x + 6) \][/tex]
4. Simplify the equation to slope-intercept form (if necessary):
Distribute [tex]\(-\frac{5}{3}\)[/tex]:
[tex]\[ y - 4 = -\frac{5}{3} x - 10 \][/tex]
Add [tex]\(4\)[/tex] to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[ y = -\frac{5}{3}x - 6 \][/tex]
5. Choose the correct answer:
From the given options:
[tex]\[ y = -\frac{5}{3}x - 6 \][/tex]
Matches exactly with the equation we derived. Thus, the correct answer is:
[tex]\[ y = -\frac{5}{3}x - 6 \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{3} \][/tex]
