x:4, 7, 10, 13
y:6, 10.5, 15, 19.5
determine whether y varies directly with x if so find the constant of variation k and write the equation
To determine if y varies directly with x, we need to check if the ratio of y to x is constant for all the given values. Let's calculate the ratio for each pair of values:
For the first pair (x = 4, y = 6):
y/x = 6/4 = 1.5
For the second pair (x = 7, y = 10.5):
y/x = 10.5/7 = 1.5
For the third pair (x = 10, y = 15):
y/x = 15/10 = 1.5
For the fourth pair (x = 13, y = 19.5):
y/x = 19.5/13 = 1.5
Since the ratio of y to x is constant (1.5) for all the values, we can conclude that y varies directly with x.
To find the constant of variation (k), we can choose any pair of values and calculate the ratio again. Let's choose the first pair:
k = y/x = 6/4 = 1.5
Therefore, the constant of variation (k) is 1.5.
The equation for direct variation between x and y is y = kx. Substituting the value of k:
y = 1.5x
To determine if y varies directly with x, we need to check if the ratio of y to x remains constant. Let's find the ratio for each pair of values:
For the first pair (4, 6), the ratio is 6/4 = 1.5.
For the second pair (7, 10.5), the ratio is 10.5/7 = 1.5.
For the third pair (10, 15), the ratio is 15/10 = 1.5.
For the fourth pair (13, 19.5), the ratio is 19.5/13 = 1.5.
Since the ratio of y to x is constant at 1.5 for all pairs, we can conclude that y varies directly with x.
To find the constant of variation (k), we can choose any pair of values and divide y by x. Let's use the first pair (4, 6):
k = y / x = 6 / 4 = 1.5
Therefore, the constant of variation is 1.5.
The equation that represents the direct variation relationship between x and y is:
y = k * x
Substituting the value of k, the equation is:
y = 1.5 * x
To determine whether y varies directly with x, we need to determine if there is a constant ratio between y and x. If there is a constant ratio, then y can be written as y = kx, where k is the constant of variation.
We can test if y varies directly with x by calculating the ratios of corresponding values of y and x and checking if they are equal.
Let's calculate the ratios using the given values:
Ratio 1: y₁ / x₁ = 6 / 4 = 1.5
Ratio 2: y₂ / x₂ = 10.5 / 7 = 1.5
Ratio 3: y₃ / x₃ = 15 / 10 = 1.5
Ratio 4: y₄ / x₄ = 19.5 / 13 = 1.5
From these calculations, we can see that all the ratios are equal to 1.5. Hence, y varies directly with x.
To find the constant of variation (k), we can use any of the ratios. Let's use Ratio 1:
k = y₁ / x₁ = 6 / 4 = 1.5
So, the constant of variation (k) is 1.5.
Now we can write the equation:
y = kx
Substituting the value of k:
y = 1.5x
Therefore, the equation of the direct variation is y = 1.5x.