determine whether y varies directly with x if so find the constant of variation k and write the equation.
X Y
6 24
18 72
54 216
162 648
To determine if y varies directly with x, we can check if the ratio between y and x stays constant.
Let's calculate the ratio between y and x for each pair of values:
For the first pair (x = 6, y = 24), the ratio is 24/6 = 4.
For the second pair (x = 18, y = 72), the ratio is 72/18 = 4.
For the third pair (x = 54, y = 216), the ratio is 216/54 = 4.
For the fourth pair (x = 162, y = 648), the ratio is 648/162 = 4.
Since the ratio between y and x is constant (4), we can conclude that y varies directly with x.
The constant of variation (k) is then equal to this ratio, which is 4.
The equation that relates y and x is given by y = kx. In this case, the equation is y = 4x.
To determine whether y varies directly with x, we need to check if the ratios of y to x are the same for all given points. Let's calculate the ratios for the given points:
For the first point (6, 24):
24 / 6 = 4
For the second point (18, 72):
72 / 18 = 4
For the third point (54, 216):
216 / 54 = 4
For the fourth point (162, 648):
648 / 162 = 4
Since the ratios are the same for all points, we conclude that y varies directly with x.
To find the constant of variation, k, we can choose any of the ratios. Let's use the first ratio:
k = 4
To write the equation, we can use the form y = kx. Substituting the value of k, we have:
y = 4x