Determine whether y varies directly with x. If so, find the constant of variation k and write the equation.
X: 6, 18, 54, 162
Y: 24, 72, 216, 648
To determine whether y varies directly with x, we can check if the ratio of y to x is constant. Let's calculate the ratio for each pair of corresponding values of x and y:
For x = 6, y = 24: y/x = 24/6 = 4
For x = 18, y = 72: y/x = 72/18 = 4
For x = 54, y = 216: y/x = 216/54 = 4
For x = 162, y = 648: y/x = 648/162 = 4
Since the ratio of y to x is constant and equal to 4, we can conclude that y varies directly with x.
The constant of variation, k, is equal to the ratio of y to x. In this case, k = 4.
The equation of variation is y = kx. Substituting k = 4, we get:
y = 4x
To determine whether y varies directly with x, we need to check if the ratio of y to x is constant for all the given values.
First, we can calculate the ratios for each pair of x and y values:
Ratio for (6, 24): 24/6 = 4
Ratio for (18, 72): 72/18 = 4
Ratio for (54, 216): 216/54 = 4
Ratio for (162, 648): 648/162 = 4
Since the ratios are all equal to 4, we can conclude that y varies directly with x.
The constant of variation (k) is the constant ratio between y and x. In this case, the constant of variation is 4.
To write the equation, we use the formula y = kx, where k represents the constant of variation. Substituting the value of k, the equation becomes:
y = 4x