Determine whether y varies directly with x. If so, find the constant of variation k and write the equation
X Y
7 11
8 13
9 15
10 17
If so, then y=kx
in other words, y/x = k is constant
so, does 11/7 = 13/8 ?? No.
In fact, y = 2x-3
so no, not direct variation
Ah, the magical world of variation. Let me put on my math clown wig and get to work!
To determine whether y varies directly with x, we need to check if the ratio of y to x is always the same. So, let's crunch some numbers together.
When x is 7, y is 11. The ratio of y to x is 11/7, which simplifies to 1.57 (approximately).
When x is 8, y is 13. The ratio of y to x is 13/8, which simplifies to 1.63 (approximately).
When x is 9, y is 15. The ratio of y to x is 15/9, which simplifies to 1.67 (approximately).
When x is 10, y is 17. The ratio of y to x is 17/10, which simplifies to 1.7 (approximately).
The ratios of y to x are not the same, my friend. So, sadly, y does not vary directly with x. But fret not, my friend, for I shall still entertain you with some humor!
Why did the mathematician get a job at the circus? Because he always found a way to solve even the most "problematic" situations! Keep smiling, my friend!
To determine whether y varies directly with x, we need to check if the ratio of y to x remains constant throughout the data set.
By examining the data given, we can see that the ratio of y to x is always 2/1 or 2.
Therefore, we can conclude that y varies directly with x.
To find the constant of variation (k), we can take any pair of x and y values and divide y by x.
Let's use the first pair (7, 11):
k = y / x
k = 11 / 7
To simplify, we can express k as k = 11/7 or k ≈ 1.57.
The equation that relates y and x in this scenario is:
y = kx
Substituting the value of k, we have:
y = 1.57x
To determine whether y varies directly with x, we need to check if the ratio between y and x remains constant. Let's find this out:
First, let's find the ratio between y and x for the given data:
For the first pair (x=7, y=11), the ratio is 11/7.
For the second pair (x=8, y=13), the ratio is 13/8.
For the third pair (x=9, y=15), the ratio is 15/9.
For the fourth pair (x=10, y=17), the ratio is 17/10.
If the ratio between y and x is the same for all pairs, then y varies directly with x.
Let's calculate the ratios:
11/7 = 1.5714
13/8 = 1.625
15/9 = 1.6667
17/10 = 1.7
As we can see, the ratios are not exactly the same. Hence, y does not vary directly with x.
Since y does not vary directly with x, we cannot find the constant of variation (k) and write the equation in the form of y = kx.