Question

A line passes through the points (-9,-3) and (6,2) what points lie on the same line
12,4
15,5
-4,2
4,0
7,3
-3,-1

Answer 1

To determine which of the given points lie on the same line defined by the points (-9, -3) and (6, 2), we first need to find the equation of the line.

1. **Calculate the slope (m):**
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Plugging in the points (-9, -3) and (6, 2):
\[
m = \frac{2 - (-3)}{6 - (-9)} = \frac{2 + 3}{6 + 9} = \frac{5}{15} = \frac{1}{3}
\]

2. **Write the equation of the line:**
We can use the point-slope form of a line with one of the points, say (-9, -3):
\[
y - y_1 = m(x - x_1)
\]
Substituting in the values:
\[
y - (-3) = \frac{1}{3}(x - (-9))
\]
Simplifying:
\[
y + 3 = \frac{1}{3}(x + 9)
\]
\[
y + 3 = \frac{1}{3}x + 3
\]
\[
y = \frac{1}{3}x + 3 - 3
\]
\[
y = \frac{1}{3}x
\]

3. **Check which points satisfy the equation \( y = \frac{1}{3}x \):**

We will now check each of the given points:

- **(12, 4)**: \( y = \frac{1}{3} \times 12 = 4 \) (Yes)
- **(15, 5)**: \( y = \frac{1}{3} \times 15 = 5 \) (Yes)
- **(-4, 2)**: \( y = \frac{1}{3} \times -4 = -\frac{4}{3} \) (No)
- **(4, 0)**: \( y = \frac{1}{3} \times 4 = \frac{4}{3} \) (No)
- **(7, 3)**: \( y = \frac{1}{3} \times 7 = \frac{7}{3} \) (No)
- **(-3, -1)**: \( y = \frac{1}{3} \times -3 = -1 \) (Yes)

4. **Conclusion:**
The points that lie on the same line are:
- (12, 4)
- (15, 5)
- (-3, -1)