Question

Points (0,2) beginning point it the race and ends at (8,10)
Several other points (2,4),(5,7),(7,8)
y =

Answer 1

To find the equation of the line connecting the points (0, 2) and (8, 10), we can use the formula for the slope \( m \) and the equation of a line.

1. Calculate the slope \( m \):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 2}{8 - 0} = \frac{8}{8} = 1
\]

2. Use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
We can use point (0, 2) as \( (x_1, y_1) \):
\[
y - 2 = 1(x - 0) \quad \Rightarrow \quad y - 2 = x
\]
Rearranging this gives:
\[
y = x + 2
\]

So the equation of the line from (0, 2) to (8, 10) is:
\[
y = x + 2
\]

Next, let's verify if the other points (2, 4), (5, 7), and (7, 8) are on the line:

- For (2, 4):
\[
y = 2 + 2 = 4 \quad \text{(point lies on the line)}
\]

- For (5, 7):
\[
y = 5 + 2 = 7 \quad \text{(point lies on the line)}
\]

- For (7, 8):
\[
y = 7 + 2 = 9 \quad \text{(this point does not lie on the line, since we get 9 instead of 8)}
\]

In summary:
- The equation of the line connecting (0, 2) and (8, 10) is \( y = x + 2 \).
- The points (2, 4) and (5, 7) lie on the line, while (7, 8) does not.