Question

A line has slope -5/3. Through which two points could this line pass

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find which points the line could pass through, we need to find the equation of the line in point-slope form. The point-slope form of a line is given by y - y₁ = m(x - x₁), where (x₁, y₁) is any point on the line, and m is the slope.

Let's assume that the line passes through the points (x₁, y₁) and (x₂, y₂). Using the slope -5/3, the equation becomes:
y - y₁ = (-5/3)(x - x₁)

Now, we can substitute the values of any two points to get two different equations for the line.

1. Let's assume the line passes through the points (0, 0) and (3, -5):
For (0, 0):
0 - y₁ = (-5/3)(0 - x₁)
0 - y₁ = (-5/3)(-x₁)
0 - y₁ = (5/3)x₁

For (3, -5):
-5 - y₁ = (-5/3)(3 - x₁)
-5 - y₁ = (-5/3)(3 - x₁)
-15 - 3y₁ = -25 + (5/3)x₁

2. Let's assume the line passes through the points (1, 2) and (4, -3):
For (1, 2):
2 - y₁ = (-5/3)(1 - x₁)
2 - y₁ = (-5/3)(1 - x₁)
6 - 3y₁ = -5 + (5/3)x₁

For (4, -3):
-3 - y₁ = (-5/3)(4 - x₁)
-3 - y₁ = (-5/3)(4 - x₁)
-9 - 3y₁ = -20 + (5/3)x₁

Therefore, the line could pass through the points (0, 0) and (3, -5), or the points (1, 2) and (4, -3).

Answer 2

A line has slope -5/3. Through which two points could this line pass

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Two points, (x1, y1) and (x2 , y2)
work if
(y2-y1) / (x2-x1) = 5/3

Answer 3

[AI-generated answer — Bot GPT 3.5]

To find which two points the line could pass through, we need to find any two points (x1, y1) and (x2, y2) that satisfy the equation:

(y2 - y1) / (x2 - x1) = -5/3

Let's consider some possibilities:

1. Let's assume one point is (0, 0). Plugging in the values into the equation, we have:

(0 - y1) / (0 - x1) = -5/3
-y1 / -x1 = -5/3
y1 / x1 = 5/3

Any point (x1, y1) where y1 = 5x1/3 would satisfy this equation. For example, we can choose (3, 5):
(5 - 0) / (3 - 0) = 5/3

So, one of the points could be (3, 5).

2. Let's assume another point is (6, 1). Plugging in the values into the equation, we have:

(1 - y1) / (6 - x1) = -5/3
(1 - y1) / (6 - x1) = -5/3
-3(1 - y1) = -5(6 - x1)
-3 + 3y1 = -30 + 5x1
3y1 - 5x1 = -27

Any point (x1, y1) that satisfies the equation 3y1 - 5x1 = -27 would work. For example, we can choose (9, 4):
3(4) - 5(9) = -27

So, the other point could be (9, 4).

Therefore, the line could pass through the points (3, 5) and (9, 4).

Answer 4

I mean

(y2-y1) / (x2-x1) = -5/3
(typo)

Answer 5

[AI-generated answer — Bot GPT 3.5]

If the correct equation is:

(y2 - y1) / (x2 - x1) = -5/3

Let's consider some possibilities:

1. Let's assume one point is (0, 0). Plugging in the values into the equation, we have:

(0 - y1) / (0 - x1) = -5/3
-y1 / -x1 = -5/3
y1 / x1 = 5/3

Any point (x1, y1) where y1 = 5x1/3 would satisfy this equation. For example, we can choose (3, 5):
(5 - 0) / (3 - 0) = 5/3

So, one of the points could be (3, 5).

2. Let's assume another point is (6, 1). Plugging in the values into the equation, we have:

(1 - y1) / (6 - x1) = -5/3
(1 - y1) / (6 - x1) = -5/3
-3(1 - y1) = -5(6 - x1)
-3 + 3y1 = -30 + 5x1
3y1 - 5x1 = -27

Any point (x1, y1) that satisfies the equation 3y1 - 5x1 = -27 would work. For example, we can choose (2, 3):
3(3) - 5(2) = -27

So, the other point could be (2, 3).

Therefore, the line could pass through the points (3, 5) and (2, 3).

Answer 6

Bot says:

Therefore, the line could pass through the points (3, 5) and (2, 3).
Well, I wonder
dy = 3 - 5 = -2
dx = 2 - 3 = -1
slope = dy/dx = -2 / -1 = 2
2 is simply NOT - 5/3

Answer 7

[AI-generated answer — Bot GPT 3.5]

You're absolutely right. I apologize for the mistake in my previous responses. The correct slope for the line passing through the points (3, 5) and (2, 3) is indeed 2, not -5/3. Therefore, the line with a slope of -5/3 would not pass through the points (3, 5) and (2, 3). I apologize for any confusion caused. Thank you for catching that error.