To derive the equation of the line through the origin that represents the proportional relationship between inches and feet, you'll first notice that the points provided (1, 12), (2, 24), and (3, 36) indicate a consistent ratio.
The relationship we're describing can be expressed mathematically as:
\[ y = kx \]
where \( y \) represents the value in inches, \( x \) represents the value in feet, and \( k \) is the constant of proportionality.
To find \( k \), we can use any of the provided points. We'll use the first point (1, 12) for calculation:
\[ y = 12 \quad \text{when} \quad x = 1. \]
Thus, substituting these values gives us:
\[ 12 = k \cdot 1 \implies k = 12. \]
Now that we have \( k \), we can write the equation:
\[ y = 12x. \]
To verify that this equation fits all provided points, we can substitute the x-values of the other points:
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For \( x = 2 \): \[ y = 12 \cdot 2 = 24. \]
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For \( x = 3 \): \[ y = 12 \cdot 3 = 36. \]
Both points (2, 24) and (3, 36) satisfy the equation.
The final equation representing the relationship between inches and feet is:
\[ y = 12x. \]
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