To derive the equation of the line through the origin (0,0) using the points given (12,1), (24,2), and (36,3), we start by identifying the constants involved in the proportional relationship.
In this case, we can observe that the points represent the relationship between inches and feet, where:
- 12 inches = 1 foot
- 24 inches = 2 feet
- 36 inches = 3 feet
To find the relationship, we can express it in terms of the same variable. The relationship can be summarized as:
\[ \text{Feet} = k \cdot \text{Inches} \]
where \( k \) is the constant of proportionality.
From the points provided:
- For (12,1): \( 1 = k \cdot 12 \) \(\Rightarrow k = \frac{1}{12}\)
- For (24,2): \( 2 = k \cdot 24 \) \(\Rightarrow k = \frac{2}{24} = \frac{1}{12}\)
- For (36,3): \( 3 = k \cdot 36 \) \(\Rightarrow k = \frac{3}{36} = \frac{1}{12}\)
In each case, we have confirmed that \( k = \frac{1}{12} \).
Now, substituting \( k \) back into the equation gives us:
\[ \text{Feet} = \frac{1}{12} \cdot \text{Inches} \]
Thus, the equation of the line through the origin representing the proportional relationship between inches and feet is:
\[ y = \frac{1}{12}x \]
Where \( y \) is the measurement in feet, and \( x \) is the measurement in inches.
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