For the data in the table, does y vary directly with x? If it does, write an equation for the direct variation.
(8,11)
(16,22)
(24,33)
Yes, y varies directly with x. The equation for the direct variation is y = (11/8)x.
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 first data point (8, 11):
Ratio = y / x = 11 / 8 = 1.375
For the second data point (16, 22):
Ratio = y / x = 22 / 16 = 1.375
For the third data point (24, 33):
Ratio = y / x = 33 / 24 = 1.375
Since the ratio of y to x remains constant (1.375) for all data points, we can conclude that y varies directly with x.
For direct variation, the equation is given by:
y = kx
To find the value of k, we can use any of the data points. Let's choose the first data point (8, 11):
11 = k * 8
Solving for k:
k = 11 / 8 = 1.375
Therefore, the equation for the direct variation is:
y = 1.375x
To determine if y varies directly with x, we need to check if the ratio between y-values and x-values is constant. Let's find the ratios for the given data points:
For the first data point (8,11):
y/x = 11/8 = 1.375
For the second data point (16,22):
y/x = 22/16 = 1.375
For the third data point (24,33):
y/x = 33/24 = 1.375
The ratios are the same for all data points, which means that y does vary directly with x.
To write the equation for direct variation, we use the formula y = kx, where k is the constant of variation. Since the ratios are all equal to 1.375, we can choose any of the data points to calculate k. Let's use the first data point (8,11):
11 = k * 8
To solve for k, divide both sides of the equation by 8:
k = 11/8 = 1.375
Therefore, the equation for direct variation is y = 1.375x.