write the equation of a line through (4,1) and:
A. parallel
B. perpendicular to the line through (7,3) and (5,-1).
=Express the equation in 5 different forms.
please help!!
A. The line through (7,3) and (5,-1)has slope
m = (y2-y1)/(x2-x1) = (-4)/(-2) = 2
To pass through (4,1) and be parallel to the given line, the equation must be
(y-1)/(x-4) = 2
y - 1 = 2x -8
y = 2x -7
I don't see why they need 5 variations of that. I'll leave that up to you.
One could be 2x -y = 7
B) To be perpendicular to the given line, the slope must be m = -1/2.
Proceed as in part A.
A. To find the equation of a line parallel to another line, we use the same slope. The slope of the line through (7,3) and (5,-1) is given by:
m = (change in y) / (change in x) = (-1 - 3) / (5 - 7) = -4 / -2 = 2
Now, we can use the point-slope form of the equation:
y - y₁ = m(x - x₁),
where (x₁, y₁) is the given point (4,1). Substituting the values:
y - 1 = 2(x - 4).
Now, we can express this equation in different forms:
1. Slope-intercept form: y = 2x - 7
2. Standard form: 2x - y = 7
3. Point-slope form: y - 1 = 2(x - 4)
4. Intercept form: x/7 + y/2 = 1
5. General form: 2x - y - 7 = 0
B. For a line perpendicular to another line, the slopes are negative reciprocals of each other. The slope of the line through (7,3) and (5,-1) is 2, so the slope of the perpendicular line is -1/2.
Using the point-slope form again, we have:
y - y₁ = m(x - x₁),
where (x₁, y₁) is the given point (4,1). Substituting the values:
y - 1 = (-1/2)(x - 4).
Now, expressing this equation in different forms:
1. Slope-intercept form: y = (-1/2)x + 3
2. Standard form: x + 2y = 7
3. Point-slope form: y - 1 = (-1/2)(x - 4)
4. Intercept form: x/7 + y/3 = 1
5. General form: x + 2y - 7 = 0
I hope this helps, but if not, I'll juggle some numbers for you!
A. To find the equation of a line parallel to another line, we need to use the same slope. The line passing through (7,3) and (5,-1) has the slope:
m = (change in y) / (change in x)
= (-1 - 3) / (5 - 7)
= -4 / -2
= 2
Now, we can use the point-slope form of the equation to find the equation of the line parallel to it and passing through the point (4,1):
y - y₁ = m(x - x₁)
y - 1 = 2(x - 4)
Expanding the equation:
y - 1 = 2x - 8
Now, let's express this equation in different forms:
1. Slope-intercept form (y = mx + b):
y = 2x - 7
2. Point-slope form (y - y₁ = m(x - x₁)):
y - 1 = 2(x - 4)
3. Standard form (Ax + By = C):
-2x + y = -7
4. General form (Ax + By + C = 0):
2x - y + 7 = 0
5. Intercept form (x / a + y / b = 1):
x / 7 + y / (-1) = 1
B. To find the line perpendicular to the line passing through (7,3) and (5,-1), we need to find the negative reciprocal of its slope. The slope of the given line is 2, so the negative reciprocal is -1/2.
Using the point-slope form as before, we can find the equation of the line perpendicular to it and passing through the point (4,1):
y - y₁ = m(x - x₁)
y - 1 = (-1/2)(x - 4)
Expanding the equation:
y - 1 = -1/2x + 2
Now, let's express this equation in different forms:
1. Slope-intercept form (y = mx + b):
y = -1/2x + 3/2
2. Point-slope form (y - y₁ = m(x - x₁)):
y - 1 = -1/2(x - 4)
3. Standard form (Ax + By = C):
x + 2y = 3
4. General form (Ax + By + C = 0):
x + 2y - 3 = 0
5. Intercept form (x / a + y / b = 1):
x / 3 + y / (3/2) = 1
To find the equation of a line, we can use the point-slope form, which is given by:
y - y1 = m(x - x1),
where (x1, y1) is a point on the line and m is the slope of the line.
A. For a line parallel to another line, the slopes are equal. So, to find the equation of a line parallel to the line passing through points (7,3) and (5,-1), we first need to find the slope of that line.
Slope (m) = (change in y) / (change in x)
= (-1 - 3) / (5 - 7)
= -4 / -2
= 2.
Now, we have the slope (m = 2). Let's use the point-slope form with the given point (4,1):
y - 1 = 2(x - 4).
To express this equation in different forms, we can simplify it:
1. Slope-intercept form (y = mx + b):
y - 1 = 2x - 8
y = 2x - 7
2. Standard form (Ax + By = C):
-2x + y = -7
3. General form (Ax + By + C = 0):
-2x + y + 7 = 0
4. Point-slope form (y - y1 = m(x - x1)):
y - 1 = 2(x - 4)
5. Intercept form (x/a + y/b = 1):
x/(-7/2) + y/1 = 1
x/(-7/2) + y = 1
B. To find the equation of a line perpendicular to the line passing through (7,3) and (5,-1), we need to find the negative reciprocal of the slope (-1/m) of that line.
Slope (m) = (change in y) / (change in x)
= (-1 - 3) / (5 - 7)
= -4 / -2
= 2.
Negative reciprocal: -1/2.
Using the point-slope form with the point (4,1):
y - 1 = (-1/2)(x - 4).
Now, let's express this equation in different forms:
1. Slope-intercept form (y = mx + b):
y - 1 = (-1/2)x + 2
y = (-1/2)x + 3
2. Standard form (Ax + By = C):
x + 2y = 6
3. General form (Ax + By + C = 0):
x + 2y - 6 = 0
4. Point-slope form (y - y1 = m(x - x1)):
y - 1 = (-1/2)(x - 4)
5. Intercept form (x/a + y/b = 1):
x/6 + y/3 = 1
These are the equations of the line parallel and perpendicular to the given line, expressed in five different forms each.