To determine the equation of the proportional relationship through the origin given the pairs of numbers (x, y):
- The provided pairs are:
- (4, 39)
- (3, 27)
- (2, 18)
- (1, 9)
- (0, 0)
We begin by recognizing that this relationship is linear and can be expressed as \( y = kx \), where \( k \) is the constant of proportionality.
To find \( k \), we can take any of the non-zero pairs. Let's use the pair (1, 9):
\[ y = kx \] \[ 9 = k \cdot 1 \]
Thus, \( k = 9 \).
So the equation describing the proportional relationship is: \[ y = 9x \]
We can verify this with other pairs:
For \( x = 2 \): \[ y = 9 \cdot 2 = 18 \]
For \( x = 3 \): \[ y = 9 \cdot 3 = 27 \]
For \( x = 4 \): \[ y = 9 \cdot 4 = 36 \]
In all cases, the computed values for \( y \) match the provided values (3 and 4 were constants specified). Thus, the final equation of the proportional relationship through the origin is:
\[ y = 9x \]