The line through (2,-3) and parallel to 2x+5y=3.
The line through (2,-3) and perpendicular to 2x+5y=3.
I just can't remember how to solve either of these kinds of problems and i can't find my notes on the subject.
The slope "m" of the first line must be -2/5. You get that by rewriting
2x + 5y = 3 as
y = -(2/5)x + 3/5
The slope "m" of the second line must be 5/2, because the product of the slopes of the two lines must be -1.
When you know the slope m and the point x1, y1 that the line goes through, the equation of the line can be written
(y - y1)/(x - x1) = m
To find the equation of a line parallel or perpendicular to a given line through a specific point, you can follow these steps:
1. Understand the given equation: To determine lines parallel or perpendicular to a given line, you need to analyze the slope (m) of the given line.
2. Find the slope (m) of the given line: Rearrange the given equation of the line into slope-intercept form (y = mx + b), where m represents the slope. In this case, the given line is 2x + 5y = 3. To express it in slope-intercept form, solve for y:
2x + 5y = 3
5y = -2x + 3
y = (-2/5)x + 3/5
Now, you can see that the slope (m) of the given line is -2/5.
3. Determine the slope of the line you are looking for:
- For a line parallel to the given line, the slope will be the same as the original line. In this case, it will also be -2/5.
- For a line perpendicular to the given line, the slope will be the negative reciprocal of the original line's slope. To find the negative reciprocal, flip the fraction and change its sign. Thus, the slope of the perpendicular line will be 5/2.
4. Use the point-slope form to find the equation of the line: Now that you have the slope (m) and a point (x1, y1) through which the line passes, you can use the point-slope form to find the equation of the line. The point-slope form is:
y - y1 = m(x - x1)
a) For a line parallel to the given line:
Given point: (2, -3) and slope (m) = -2/5
Substitute the values into the point-slope form:
y - (-3) = (-2/5)(x - 2)
Simplify and rearrange to obtain the equation in slope-intercept form (y = mx + b):
y = (-2/5)x + (2/5) - 3
y = (-2/5)x - (13/5)
b) For a line perpendicular to the given line:
Given point: (2, -3) and slope (m) = 5/2
Substitute the values into the point-slope form:
y - (-3) = (5/2)(x - 2)
Simplify and rearrange to obtain the equation in slope-intercept form (y = mx + b):
y = (5/2)x -5/2
Therefore, the equation of the line parallel to 2x + 5y = 3 and passing through (2, -3) is y = (-2/5)x - (13/5), and the equation of the line perpendicular to 2x + 5y = 3 and passing through (2, -3) is y = (5/2)x - 5/2.