Given an equation of a line, how do I find equations for lines parallel to it going through specified points Simplifying the equations into slope-intercept form.
1. y= -2x -4; (1, 3) how do I write the equation of a line parallel to the given line but passing through the given point
2. y= 3x + 3; (1,1) how do I write the equation of a line perpendicular to the given line but passing through the given point
It must have the same slope: -2
Now you have a point and a slope, so the line is
y-3 = -2(x-1)
same idea, but you want the perpendicular slope, which is the negative reciprocal of the line's slope: -1/3
y-1 = -1/3(x-1)
review the various forms of the equation of a line:
standard
point-slope
two-point
slope-intercept
1. Y = -2x - 4. (1,3).
Parallel lines have equal slopes:
m1 = m2 = -2.
Y = mx + b = 3
-2*1 + b = 3
b = 3+2 = 5
Eq: Y = -2x + 5
2. Y = 3x+3. (1,1).
The slope of the line is equal to the negat1ive reciprocal of the slope of the given line.
m1 = 3
m2 = -1/3.
Y = mx + b = 1
(-1/3)*1 + b = 1
b = 1 + 1/3 = 1 1/3 = 4/3
Eq: Y = (-1/3)x + 4/3
To find the equations for lines parallel or perpendicular to a given line and passing through specified points, you need to understand the concept of slope.
1. To find the equation of a line parallel to the given line but passing through the given point:
- The parallel lines will have the same slope as the given line.
- Given line: y = -2x - 4
- The slope of the given line is -2.
- Since the desired line is parallel, it will also have a slope of -2.
- Use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
- Substitute the values: (x1, y1) = (1, 3) and m = -2
- The equation becomes: y - 3 = -2(x - 1)
- Simplify the equation: y - 3 = -2x + 2
- Convert the equation to slope-intercept form (y = mx + b): y = -2x + 5
Therefore, the equation of the line parallel to y = -2x - 4 and passing through (1, 3) is y = -2x + 5.
2. To find the equation of a line perpendicular to the given line but passing through the given point:
- The perpendicular lines will have a slope that is the negative reciprocal of the slope of the given line.
- Given line: y = 3x + 3
- The slope of the given line is 3.
- The negative reciprocal of 3 is -1/3.
- Use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
- Substitute the values: (x1, y1) = (1, 1) and m = -1/3
- The equation becomes: y - 1 = (-1/3)(x - 1)
- Simplify the equation: y - 1 = -1/3x + 1/3
- Convert the equation to slope-intercept form (y = mx + b): y = -1/3x + 4/3
Therefore, the equation of the line perpendicular to y = 3x + 3 and passing through (1, 1) is y = -1/3x + 4/3.