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Sagot :
To find the equation of a line that is perpendicular to the given line [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex] and passes through the point [tex]\((-2, -2)\)[/tex], follow these steps:
1. Determine the slope of the given line:
The given line is in point-slope form, [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope. From the equation [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex].
2. Find the slope of the perpendicular line:
Lines that are perpendicular have slopes that are negative reciprocals of each other. So, if the slope of the given line is [tex]\( -\frac{2}{3} \)[/tex], the slope of the perpendicular line is [tex]\( \frac{3}{2} \)[/tex].
3. Use the point-slope form to find the y-intercept:
We know the slope of the perpendicular line is [tex]\( \frac{3}{2} \)[/tex], and it passes through the point [tex]\((-2, -2)\)[/tex]. We can use the point-slope form of the equation:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( x_1 = -2 \)[/tex], [tex]\( y_1 = -2 \)[/tex], and [tex]\( m = \frac{3}{2} \)[/tex]. Plugging these values in:
[tex]\[ y - (-2) = \frac{3}{2}(x - (-2)) \][/tex]
Simplifying this, we get:
[tex]\[ y + 2 = \frac{3}{2}(x + 2) \][/tex]
Distribute the slope:
[tex]\[ y + 2 = \frac{3}{2}x + \frac{3}{2} \cdot 2 \][/tex]
[tex]\[ y + 2 = \frac{3}{2}x + 3 \][/tex]
4. Isolate [tex]\( y \)[/tex] to put the equation in slope-intercept form ( [tex]\( y = mx + b \)[/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 [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex] and passes through the point [tex]\((-2, -2)\)[/tex] is:
[tex]\[ y = \frac{3}{2}x + 1 \][/tex]
Therefore, the correct choice is:
[tex]\[ y = \frac{3}{2}x + 1 \][/tex]
1. Determine the slope of the given line:
The given line is in point-slope form, [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope. From the equation [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex].
2. Find the slope of the perpendicular line:
Lines that are perpendicular have slopes that are negative reciprocals of each other. So, if the slope of the given line is [tex]\( -\frac{2}{3} \)[/tex], the slope of the perpendicular line is [tex]\( \frac{3}{2} \)[/tex].
3. Use the point-slope form to find the y-intercept:
We know the slope of the perpendicular line is [tex]\( \frac{3}{2} \)[/tex], and it passes through the point [tex]\((-2, -2)\)[/tex]. We can use the point-slope form of the equation:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( x_1 = -2 \)[/tex], [tex]\( y_1 = -2 \)[/tex], and [tex]\( m = \frac{3}{2} \)[/tex]. Plugging these values in:
[tex]\[ y - (-2) = \frac{3}{2}(x - (-2)) \][/tex]
Simplifying this, we get:
[tex]\[ y + 2 = \frac{3}{2}(x + 2) \][/tex]
Distribute the slope:
[tex]\[ y + 2 = \frac{3}{2}x + \frac{3}{2} \cdot 2 \][/tex]
[tex]\[ y + 2 = \frac{3}{2}x + 3 \][/tex]
4. Isolate [tex]\( y \)[/tex] to put the equation in slope-intercept form ( [tex]\( y = mx + b \)[/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 [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex] and passes through the point [tex]\((-2, -2)\)[/tex] is:
[tex]\[ y = \frac{3}{2}x + 1 \][/tex]
Therefore, the correct choice is:
[tex]\[ y = \frac{3}{2}x + 1 \][/tex]
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