Question

A line passes through the points (-9,-3) and (6,2) what points lie on the same line

Answer 1

To find other points that lie on the line passing through the points \((-9, -3)\) and \((6, 2)\), we first need to determine the slope of the line.

The slope \(m\) is calculated using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

In this case, let \((x_1, y_1) = (-9, -3)\) and \((x_2, y_2) = (6, 2)\):

\[
m = \frac{2 - (-3)}{6 - (-9)} = \frac{2 + 3}{6 + 9} = \frac{5}{15} = \frac{1}{3}
\]

Now that we know the slope is \(\frac{1}{3}\), we can use the point-slope form of the equation of a line:

\[
y - y_1 = m(x - x_1)
\]

Using point \((-9, -3)\):

\[
y - (-3) = \frac{1}{3}(x - (-9))
\]

This simplifies to:

\[
y + 3 = \frac{1}{3}(x + 9)
\]

Now multiplying through by 3 to eliminate the fraction:

\[
3(y + 3) = x + 9
\]

Expanding gives us:

\[
3y + 9 = x + 9
\]

Subtracting 9 from both sides results in:

\[
3y = x
\]

Rearranging, we obtain the equation of the line:

\[
x - 3y = 0
\]

To find other points on this line, we can choose various values of \(x\) and calculate corresponding \(y\) values using the equation \(y = \frac{x}{3}\).

Here are a few examples:

1. If \(x = 0\):
\[
y = \frac{0}{3} = 0 \implies (0, 0)
\]

2. If \(x = 3\):
\[
y = \frac{3}{3} = 1 \implies (3, 1)
\]

3. If \(x = 6\):
\[
y = \frac{6}{3} = 2 \implies (6, 2) \text{ (already known)}
\]

4. If \(x = -3\):
\[
y = \frac{-3}{3} = -1 \implies (-3, -1)
\]

5. If \(x = -6\):
\[
y = \frac{-6}{3} = -2 \implies (-6, -2)
\]

Thus, some additional points that lie on the same line include \((0, 0)\), \((3, 1)\), \((-3, -1)\), and \((-6, -2)\). You can create more points by choosing other \(x\) values.