Given the points [tex]\((-6,3)\)[/tex] and [tex]\((3,6)\)[/tex], we will calculate the following:
1. The difference in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates (delta)^:
[tex]\[
\Delta x = x_2 - x_1 = 3 - (-6) = 9
\][/tex]
[tex]\[
\Delta y = y_2 - y_1 = 6 - 3 = 3
\][/tex]
2. The midpoint of the line segment connecting the two points:
[tex]\[
\text{Midpoint } \left(x_m, y_m\right) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\][/tex]
[tex]\[
x_m = \frac{-6 + 3}{2} = \frac{-3}{2} = -1.5
\][/tex]
[tex]\[
y_m = \frac{3 + 6}{2} = \frac{9}{2} = 4.5
\][/tex]
3. The slope [tex]\(m\)[/tex] of the line passing through the two points:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{3}{9} = \frac{1}{3} \approx 0.3333
\][/tex]
4. The distance between the two points:
[tex]\[
\text{Distance} = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\][/tex]
[tex]\[
\text{Distance} = \sqrt{9^2 + 3^2} = \sqrt{81 + 9} = \sqrt{90} \approx 9.4868
\][/tex]
5. The equation of the line in slope-intercept form [tex]\(y = mx + b\)[/tex]:
Using the point [tex]\((-6, 3)\)[/tex]:
[tex]\[
y = mx + b \Rightarrow 3 = \left(\frac{1}{3}\right)(-6) + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
3 = -2 + b \Rightarrow b = 3 + 2 = 5
\][/tex]
Thus, the various properties and calculations related to the points [tex]\((-6,3)\)[/tex] and [tex]\((3,6)\)[/tex] are:
- [tex]\(\Delta x = 9\)[/tex]
- [tex]\(\Delta y = 3\)[/tex]
- Midpoint: [tex]\((-1.5, 4.5)\)[/tex]
- Slope: [tex]\(\approx 0.3333\)[/tex]
- Distance: [tex]\(\approx 9.4868\)[/tex]
- y-intercept [tex]\(b\)[/tex] of the line: 5
