Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To find the equation of the line that is perpendicular to the line [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex] and passes through the point [tex]\((-2, -2)\)[/tex], we need to follow these steps:
1. Find the slope of the given line:
The given line is in point-slope form: [tex]\( y - y_1 = m(x - x_1) \)[/tex].
From [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex], we see that the slope [tex]\( m \)[/tex] of the given line is [tex]\( -\frac{2}{3} \)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope.
[tex]\[ \text{Slope of perpendicular line} = -\frac{1}{m} = -\frac{1}{-\frac{2}{3}} = \frac{3}{2} \][/tex]
3. Use the point-slope form to write the equation of the perpendicular line:
We will use the point [tex]\((-2, -2)\)[/tex] and the slope [tex]\(\frac{3}{2}\)[/tex].
The point-slope form of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Plugging in the point and the slope:
[tex]\[ y - (-2) = \frac{3}{2}(x - (-2)) \][/tex]
Simplifying:
[tex]\[ y + 2 = \frac{3}{2}(x + 2) \][/tex]
[tex]\[ y + 2 = \frac{3}{2}x + \frac{3}{2} \cdot 2 \][/tex]
[tex]\[ y + 2 = \frac{3}{2}x + 3 \][/tex]
4. Convert to slope-intercept form [tex]\( y = mx + b \)[/tex]:
Subtract 2 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = \frac{3}{2}x + 3 - 2 \][/tex]
[tex]\[ y = \frac{3}{2}x + 1 \][/tex]
So, the equation of the line that is perpendicular to the given line and passes through the point [tex]\((-2, -2)\)[/tex] is:
[tex]\[ y = \frac{3}{2}x + 1 \][/tex]
Hence, the correct answer is:
[tex]\[ y = \frac{3}{2}x + 1 \][/tex]
1. Find the slope of the given line:
The given line is in point-slope form: [tex]\( y - y_1 = m(x - x_1) \)[/tex].
From [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex], we see that the slope [tex]\( m \)[/tex] of the given line is [tex]\( -\frac{2}{3} \)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope.
[tex]\[ \text{Slope of perpendicular line} = -\frac{1}{m} = -\frac{1}{-\frac{2}{3}} = \frac{3}{2} \][/tex]
3. Use the point-slope form to write the equation of the perpendicular line:
We will use the point [tex]\((-2, -2)\)[/tex] and the slope [tex]\(\frac{3}{2}\)[/tex].
The point-slope form of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Plugging in the point and the slope:
[tex]\[ y - (-2) = \frac{3}{2}(x - (-2)) \][/tex]
Simplifying:
[tex]\[ y + 2 = \frac{3}{2}(x + 2) \][/tex]
[tex]\[ y + 2 = \frac{3}{2}x + \frac{3}{2} \cdot 2 \][/tex]
[tex]\[ y + 2 = \frac{3}{2}x + 3 \][/tex]
4. Convert to slope-intercept form [tex]\( y = mx + b \)[/tex]:
Subtract 2 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = \frac{3}{2}x + 3 - 2 \][/tex]
[tex]\[ y = \frac{3}{2}x + 1 \][/tex]
So, the equation of the line that is perpendicular to the given line and passes through the point [tex]\((-2, -2)\)[/tex] is:
[tex]\[ y = \frac{3}{2}x + 1 \][/tex]
Hence, the correct answer is:
[tex]\[ y = \frac{3}{2}x + 1 \][/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.