X Y
4 6.4
7 11.2
10 16
13 20.8
Determine whether y varies directly with x. If so, find the constant of variation k and write the equation.
To determine whether y varies directly with x, we need to check if the ratio of y to x is constant for all values.
Let's calculate the ratio of y to x for each pair:
For the first pair (4, 6.4):
ratio = 6.4/4 = 1.6
For the second pair (7, 11.2):
ratio = 11.2/7 = 1.6 (same as the first pair)
For the third pair (10, 16):
ratio = 16/10 = 1.6 (same as the previous pairs)
For the fourth pair (13, 20.8):
ratio = 20.8/13 = 1.6 (same as the previous pairs)
Since the ratio of y to x is constant (1.6) for all pairs, we can conclude that y varies directly with x.
To find the constant of variation (k), we can select any pair of x and y and use the formula:
k = y / x
Let's choose the first pair (4, 6.4):
k = 6.4 / 4 = 1.6
Therefore, the constant of variation is k = 1.6.
The equation for the direct variation is:
y = kx
Substituting the value of k, we get:
y = 1.6x
To determine whether y varies directly with x, we need to check if the ratio of y to x remains constant for all data points. Let's calculate the ratios for the given data:
For the first data point (4, 6.4):
y / x = 6.4 / 4 = 1.6
For the second data point (7, 11.2):
y / x = 11.2 / 7 = 1.6
For the third data point (10, 16):
y / x = 16 / 10 = 1.6
For the fourth data point (13, 20.8):
y / x = 20.8 / 13 = 1.6
Since the ratio of y to x remains constant (1.6) for all data points, we can conclude that y varies directly with x.
Now, let's find the constant of variation, k. The constant of variation represents the ratio of y to x.
Taking any of the data points, let's use the first one (4, 6.4):
k = y / x = 6.4 / 4 = 1.6
So, the constant of variation, k, is 1.6.
Finally, the equation that represents the direct variation relationship between x and y is:
y = kx
In this case, the equation is:
y = 1.6x