A line passes through the point (3,-1)
and has a slope of 2/3 Which of the
following points also lies on this line?
-
A. (9,5)****
B. (-3, 1)
C. (6,3)
D. (-9,7)
Well, well, well! Let's see here. We have a line with a slope of 2/3 passing through the point (3,-1). Now, if we want to find another point that lies on this line, we need to check which of the given options satisfies the equation of the line.
The equation for a line is y = mx + b, where m represents the slope and b is the y-intercept.
So, let's plug in the values we have. For the given line, we have the slope as 2/3, and we know that it passes through the point (3,-1). Let's substitute these values into the equation.
-1 = (2/3)(3) + b
Simplifying this equation, we get:
-1 = 2 + b
To find the value of b, we subtract 2 from both sides of the equation:
b = -1 - 2
b = -3
So, the equation for our line is y = (2/3)x - 3.
Now that we have our equation, we can check which of the given points satisfy it.
Let's try the first option, (9,5).
Is 5 = (2/3)(9) - 3?
5 = 18/3 - 3
5 = 6 - 3
5 = 3
Oops! That doesn't work. So A is out.
Let's move on to the next option, (-3,1).
Is 1 = (2/3)(-3) - 3?
1 = -6/3 - 3
1 = -2 - 3
1 = -5
Nope, that's not it. So B is out too.
Now let's give the third option, (6,3), a shot.
Is 3 = (2/3)(6) - 3?
3 = 12/3 - 3
3 = 4 - 3
3 = 1
Nope, that doesn't add up either. So C is not the right answer.
Finally, let's try the last option, (-9,7).
Is 7 = (2/3)(-9) - 3?
7 = -18/3 - 3
7 = -6 - 3
7 = -9
Well, well, well! This option satisfies the equation! The point (-9,7) lies on the line.
So, my dear friend, the correct answer is D. (-9,7). Keep those math hats on, and let's keep moving!
To determine which point lies on the line with a slope of 2/3 and passing through the point (3, -1), we can use the point-slope form of a linear equation, which is given by:
y - y1 = m(x - x1)
Where (x1, y1) represents the given point and m represents the slope.
Let's substitute the given values into the equation:
y - (-1) = (2/3)(x - 3)
Simplifying this equation gives:
y + 1 = (2/3)(x - 3)
Now, let's check which of the points satisfy this equation.
A. (9, 5):
Plugging in x = 9 and y = 5 into the equation:
5 + 1 = (2/3)(9 - 3)
6 = (2/3)(6)
6 = 4
This is not a valid solution, so point (9, 5) does not lie on the line.
B. (-3, 1):
Plugging in x = -3 and y = 1 into the equation:
1 + 1 = (2/3)(-3 - 3)
2 = (2/3)(-6)
2 = -4
This is not a valid solution, so point (-3, 1) does not lie on the line.
C. (6, 3):
Plugging in x = 6 and y = 3 into the equation:
3 + 1 = (2/3)(6 - 3)
4 = (2/3)(3)
4 = 2
This is not a valid solution, so point (6, 3) does not lie on the line.
D. (-9, 7):
Plugging in x = -9 and y = 7 into the equation:
7 + 1 = (2/3)(-9 - 3)
8 = (2/3)(-12)
8 = -8
This is not a valid solution, so point (-9, 7) does not lie on the line.
Therefore, the correct answer is A. (9, 5)
To determine which of the given points lie on the line with a slope of 2/3 passing through the point (3, -1), we can use the point-slope formula.
The point-slope formula for a line is:
y - y1 = m(x - x1)
Where (x1, y1) represents a point on the line, m represents the slope, and (x, y) represents any other point on the line.
Given:
Point on line: (3, -1)
Slope: 2/3
Let's substitute these values into the point-slope formula:
y - (-1) = (2/3)(x - 3)
Simplifying:
y + 1 = (2/3)(x - 3)
Now, let's check which of the given points satisfy this equation.
For point (9, 5):
y + 1 = (2/3)(x - 3)
5 + 1 = (2/3)(9 - 3)
6 = (2/3)(6)
6 = 4
The equation does not hold true for point (9, 5), so this point does not lie on the line.
For point (-3, 1):
y + 1 = (2/3)(x - 3)
1 + 1 = (2/3)(-3 - 3)
2 = (2/3)(-6)
2 = -4
The equation does not hold true for point (-3, 1), so this point does not lie on the line.
For point (6, 3):
y + 1 = (2/3)(x - 3)
3 + 1 = (2/3)(6 - 3)
4 = (2/3)(3)
4 = 2
The equation does not hold true for point (6, 3), so this point does not lie on the line.
For point (-9, 7):
y + 1 = (2/3)(x - 3)
7 + 1 = (2/3)(-9 - 3)
8 = (2/3)(-12)
8 = -8
The equation does not hold true for point (-9, 7), so this point does not lie on the line.
Therefore, the point (9, 5) does not lie on the line with a slope of 2/3 passing through the point (3, -1).
The correct answer is B. (-3, 1).
no, the slope between your choice of
(9,5) and (3,-1) would be
(5-(-1))/(9-3) = 6/6 = 1 and not 2/3
keep working at it.
Given: (3, -1), m = 2/3.
Eq: Y = (2/3)x - 3
None of the given choices satisfy the derived Eq.
The following points satisfy the derived Eq.:
(3, -1), (9, 3), (-3, -5), (6, 1), (-9, -9).