Determine whether y varies directly with x. If so find the constant of variation k and write the equation. X 4,7,10,13. Y 6.4,11.2,16,20.8
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 ratios for the given values:
For x = 4, y = 6.4; y/x = 6.4/4 = 1.6
For x = 7, y = 11.2; y/x = 11.2/7 = 1.6
For x = 10, y = 16; y/x = 16/10 = 1.6
For x = 13, y = 20.8; y/x = 20.8/13 ≈ 1.6
Since the ratio of y to x is the same for all values, we can conclude that y varies directly with x.
The constant of variation, k, can be found by calculating the ratio of y to x for any of the given values. Let's use the first pair, x = 4 and y = 6.4:
k = y/x = 6.4/4 = 1.6
The equation expressing the direct variation in this case is:
y = kx
Substituting k = 1.6, the equation becomes:
y = 1.6x
To determine whether y varies directly with x, we need to check if the ratio of y to x is always the same. Let's calculate the ratios for each pair of x and y values:
For the first pair (x=4, y=6.4), the ratio is y/x = 6.4/4 = 1.6.
For the second pair (x=7, y=11.2), the ratio is y/x = 11.2/7 = 1.6.
For the third pair (x=10, y=16), the ratio is y/x = 16/10 = 1.6.
For the fourth pair (x=13, y=20.8), the ratio is y/x = 20.8/13 = 1.6.
Since the ratio is the same for all pairs, 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 ratio is 1.6. Therefore, k = 1.6.
To write the equation, we can use the general form of direct variation: y = kx. Substituting the value of k, the equation becomes y = 1.6x.
To determine if y varies directly with x, we need to check if the ratios between corresponding values of x and y remain constant. In other words, if for every x, y/x is always the same value, then y varies directly with x.
Let's calculate the ratios for the given values of x and y:
For x = 4, y = 6.4:
y/x = 6.4/4 = 1.6
For x = 7, y = 11.2:
y/x = 11.2/7 = 1.6
For x = 10, y = 16:
y/x = 16/10 = 1.6
For x = 13, y = 20.8:
y/x = 20.8/13 = 1.6
As we can see, the ratio y/x is the same (1.6) for all the given values. Therefore, y varies directly with x.
Now, to find the constant of variation (k), we can take any x and y value pair and divide them. Let's use the first pair (x=4, y=6.4):
k = y/x = 6.4/4 = 1.6
The equation for the direct variation is:
y = kx
Substituting the value of k:
y = 1.6x