Given a proportional relationship between inches and feet, we can express this relationship with a linear equation in the form:
\[ y = kx \]
where \( y \) represents the values in feet, \( x \) represents the values in inches, and \( k \) is a constant representing the proportionality factor.
Step 1: Determine the Proportionality Constant \( k \)
Using the first point (1, 12):
- Here, \((x, y) = (1, 12)\)
Substituting into the equation:
\[ 12 = k(1) \]
Thus,
\[ k = 12 \]
We can check this constant with the other points.
Step 2: Check the other points
For the second point (2, 24):
- \((x, y) = (2, 24)\)
Substituting into the equation:
\[ 24 = k(2) \]
Substituting \( k = 12 \):
\[ 24 = 12(2) \]
This holds true.
For the third point (3, 36):
- \((x, y) = (3, 36)\)
Substituting into the equation:
\[ 36 = k(3) \]
Again substituting \( k = 12 \):
\[ 36 = 12(3) \]
This also holds true.
Step 3: Write the equation of the line
Since we have confirmed that the constant \( k \) is consistent across all points, we can write the equation of the line that represents the relationship between inches and feet as:
\[ y = 12x \]
This equation indicates that for every inch, there are 12 inches in a foot. This is a linear equation with a slope of 12, going through the origin (0, 0).
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