To determine the slope of the line that represents the given proportional relationship, we need to follow a systematic approach. Below is a step-by-step solution for finding the slope:
1. Identify Two Points on the Line:
Given the table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 0 & 0.3 & 0.6 & 0.9 & 1.2 \\
y & 0 & 1 & 2 & 3 & 4 \\
\hline
\end{array}
\][/tex]
We can pick any two points from this table to calculate the slope. For simplicity, let's choose the first two points:
[tex]\[
A(0, 0)
\][/tex]
[tex]\[
B(0.3, 1)
\][/tex]
2. Apply the Slope Formula:
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting in the coordinates of points A and B, we get:
[tex]\[
x_1 = 0, \quad y_1 = 0
\][/tex]
[tex]\[
x_2 = 0.3, \quad y_2 = 1
\][/tex]
[tex]\[
m = \frac{1 - 0}{0.3 - 0} = \frac{1}{0.3}
\][/tex]
3. Simplify the Expression:
Simplify the fraction to find the slope:
[tex]\[
m = \frac{1}{0.3} = \frac{1}{\frac{3}{10}} = 1 \times \frac{10}{3} = \frac{10}{3} \approx 3.333
\][/tex]
Therefore, the slope of the line that represents this relationship is approximately [tex]\( 3.333 \)[/tex].
Graphing the Line:
To graph the line, consider the points from the table:
- (0, 0)
- (0.3, 1)
- (0.6, 2)
- (0.9, 3)
- (1.2, 4)
These points can be plotted on a coordinate plane, and a straight line passing through these points will visually represent the relationship. The line should rise by approximately [tex]\( 3.333 \)[/tex] units in the y-direction for every 1 unit it runs in the x-direction.
1. Plot the Points:
- Start at the origin [tex]\((0, 0)\)[/tex].
- Move right to [tex]\( x = 0.3 \)[/tex] and up to [tex]\( y = 1 \)[/tex] to plot the point [tex]\((0.3, 1)\)[/tex].
- Continue this pattern for the remaining points [tex]\((0.6, 2)\)[/tex], [tex]\((0.9, 3)\)[/tex], and [tex]\((1.2, 4)\)[/tex].
2. Draw the Line:
Connect these points with a straight line. This line will have a slope of [tex]\( 3.333 \)[/tex] and will pass through all the points mentioned in the table.
Thus, the graph will be a straight line that confirms the proportional relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] with the slope calculated.
