Let's solve the problem step by step.
We are given two points on the line, (0, 1) and (5, 11), and we need to find the slope of the line passing through these points.
1. Identify the coordinates of the points:
- Point 1: [tex]\( (x_1, y_1) = (0, 1) \)[/tex]
- Point 2: [tex]\( (x_2, y_2) = (5, 11) \)[/tex]
2. Calculate the change in [tex]\( y \)[/tex] (Δy):
[tex]\[
\Delta y = y_2 - y_1 = 11 - 1 = 10
\][/tex]
So, the change in [tex]\( y \)[/tex] is 10.
3. Calculate the change in [tex]\( x \)[/tex] (Δx):
[tex]\[
\Delta x = x_2 - x_1 = 5 - 0 = 5
\][/tex]
So, the change in [tex]\( x \)[/tex] is 5.
4. Determine the slope (m):
The formula for the slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the values we found:
[tex]\[
m = \frac{10}{5} = 2
\][/tex]
So, the slope is 2.
Therefore, the answers to the fill-in-the-blank sections are:
- The change in [tex]\( y \)[/tex] is 10.
- The change in [tex]\( x \)[/tex] is 5.
- The slope is [tex]\( \frac{10}{5} \)[/tex].
- After simplifying, the slope is 2.
