To determine the equation of a line given points [tex]\((-3, 3)\)[/tex], [tex]\((0, 0)\)[/tex], and [tex]\((3, -3)\)[/tex] that lie on it, we can follow these steps:
1. Identify the points:
- Point 1: [tex]\((-3, 3)\)[/tex]
- Point 2: [tex]\((0, 0)\)[/tex]
- Point 3: [tex]\((3, -3)\)[/tex]
2. Calculate the slope (m) of the line:
The slope between any two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
First, finding the slope between point 1 [tex]\((-3, 3)\)[/tex] and point 2 [tex]\((0, 0)\)[/tex]:
[tex]\[
m = \frac{0 - 3}{0 - (-3)} = \frac{-3}{3} = -1
\][/tex]
Next, calculating the slope between point 2 [tex]\((0, 0)\)[/tex] and point 3 [tex]\((3, -3)\)[/tex]:
[tex]\[
m = \frac{-3 - 0}{3 - 0} = \frac{-3}{3} = -1
\][/tex]
Since both slopes are equal ([tex]\(-1\)[/tex]), it confirms all three points are collinear and lie on the same line.
3. Formulating the equation:
Using the point-slope form of a linear equation which is [tex]\(y - y_1 = m(x - x_1)\)[/tex], we can simplify this process by choosing point 2 [tex]\((0, 0)\)[/tex]. Putting the values into the equation where [tex]\( x_1 = 0 \)[/tex] and [tex]\( y_1 = 0 \)[/tex]:
[tex]\[
y - 0 = -1(x - 0)
\][/tex]
Simplifying, we get:
[tex]\[
y = -1 \cdot x
\][/tex]
Which simplifies to:
[tex]\[
y = -x
\][/tex]
So, the linear equation that passes through the points [tex]\((-3, 3)\)[/tex], [tex]\((0, 0)\)[/tex], and [tex]\((3, -3)\)[/tex] is:
[tex]\[
y = -x
\][/tex]
This equation accurately describes the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] for any point on the line.
