To derive the equation of the proportional relationship through the origin from the given points, we need to identify the constant of proportionality.
The points you provided are:
- (1, 9)
- (2, 18)
- (3, 27)
- (4, 36)
In a proportional relationship, if \( y \) is directly proportional to \( x \), then the relationship can be expressed as:
\[ y = kx \]
where \( k \) is the constant of proportionality.
We can find \( k \) using any of the points provided. Let's use the first point (1, 9):
Given \( x = 1 \) and \( y = 9 \):
\[ 9 = k(1) \] \[ k = 9 \]
Now, we can write the equation of the proportional relationship:
\[ y = 9x \]
To confirm this relationship holds for the other points:
-
For (2, 18): \[ y = 9(2) = 18 \quad \text{(correct)} \]
-
For (3, 27): \[ y = 9(3) = 27 \quad \text{(correct)} \]
-
For (4, 36): \[ y = 9(4) = 36 \quad \text{(correct)} \]
Thus, the derived equation of the proportional relationship through the origin is:
\[ y = 9x \]
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