Find the equation of the line that contains the point (-4,-6) and is parallel to the line that has the equation x+4y=5
Find the equation of the line that contains the point (9,7) and is perpendicular to the line that has the equation 2x-3y=3
y+6 = -1/4 (x+4)
y-7 = -3/2 (x-9)
OMG, thank you!
To find the equation of a line, we need to determine its slope and its y-intercept.
1. Parallel line:
The equation x + 4y = 5 can be written in slope-intercept form, y = mx + b, by solving for y:
4y = -x + 5
y = -(1/4)x + 5/4
Since the given line is parallel to the line we are trying to find, their slopes will be equal. Therefore, the slope of the line we are looking for is -1/4.
Using the point-slope form, y - y1 = m(x - x1), where (x1, y1) is the given point (-4, -6) and m is the slope:
y - (-6) = -(1/4)(x - (-4))
y + 6 = -(1/4)(x + 4)
Simplifying the equation gives us the final answer:
y + 6 = -(1/4)x - 1
y = -(1/4)x - 7
Therefore, the equation of the line that contains the point (-4, -6) and is parallel to x + 4y = 5 is y = -(1/4)x - 7.
2. Perpendicular line:
To find the equation of a line perpendicular to 2x - 3y = 3, we need to determine the negative reciprocal of its slope, as perpendicular lines have slopes that multiply to give -1.
The equation 2x - 3y = 3 can be written in slope-intercept form as follows:
3y = 2x - 3
y = (2/3)x - 1
The slope of the given line is 2/3. The negative reciprocal of 2/3 is -3/2.
Now, using the point-slope form with the point (9, 7) and the slope -3/2:
y - 7 = -3/2(x - 9)
Simplifying the equation gives us the final answer:
y - 7 = -3/2x + 27/2
y = -3/2x + 27/2 + 14/2
y = -3/2x + 41/2
Therefore, the equation of the line that contains the point (9, 7) and is perpendicular to 2x - 3y = 3 is y = -3/2x + 41/2.