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x: 4, 7, 10, 13
y: 6.4, 11.2, 16, 20.8
Determine whether y varies directly with x. If so, find the constant variation k and write the equation.
To determine whether y varies directly with x, we need to see if there is a constant ratio between the corresponding values of x and y.
First, we can calculate the ratio between consecutive values of x and y:
For the first pair of values (4, 6.4):
Ratio = y/x = 6.4/4 = 1.6
For the second pair of values (7, 11.2):
Ratio = y/x = 11.2/7 = 1.6
For the third pair of values (10, 16):
Ratio = y/x = 16/10 = 1.6
For the fourth pair of values (13, 20.8):
Ratio = y/x = 20.8/13 = 1.6
In each case, we have the same ratio of 1.6. This indicates that y varies directly with x.
To find the constant variation k, we can choose any pair of values and divide y by x. Let's use the first pair of values:
k = y/x = 6.4/4 = 1.6
Therefore, the constant variation is k = 1.6.
The equation that represents the direct variation is:
y = kx
Substituting the value of k, we have:
y = 1.6x
To determine if y varies directly with x, we need to check if the ratio of y to x is constant.
Let's calculate the ratios of y to x for each pair of values:
For x = 4, y = 6.4, the ratio y/x = 6.4/4 = 1.6
For x = 7, y = 11.2, the ratio y/x = 11.2/7 ≈ 1.6
For x = 10, y = 16, the ratio y/x = 16/10 = 1.6
For x = 13, y = 20.8, the ratio y/x = 20.8/13 ≈ 1.6
The ratio y/x is the same for each pair, which means y varies directly with x.
To find the constant of variation (k), we can choose any of the ratios calculated above since they are all the same. Let's take the ratio from the first pair:
k = y/x = 6.4/4 = 1.6
The equation representing the direct variation between x and y is:
y = kx
Replacing k with the value we found:
y = 1.6x