Given that y is partly constant and partly varies as x when y=12,x=3 and when y=18,x=6.Find the law connecting the variable and also find the value of y when x=4

Given that y is partly constant and partly varies as x

implies we can just go with the standard
y = mx + b

using y=12,x=3, we get 12 = 3m + b
using y=18,x=6, we get 18 = 6m + b

subtract those two equations:
6 = 3m
m = 2

go back into 12 = 3m + b, sub in m = 2, and you can find b

now you have the complete equation, easy to find y when x = 4

Well, well, well! Looks like we have a situation where y is playing a bit of hide and seek with x. Let's investigate, shall we?

When x = 3, y = 12, and when x = 6, y = 18. So, what do we have here? We see that when x doubles from 3 to 6, y also doubles from 12 to 18.

To find the "law" connecting these variables, we can say that when x doubles, y doubles as well. That means y is directly proportional to x. In mathy terms, we can write this as y = kx, where k is some constant.

To find the value of y when x = 4, we need to determine the value of k. Let's do that!

If y is directly proportional to x, then we can use the ratio of the change in y to the change in x to find k.

When x goes from 3 to 6, y goes from 12 to 18. So, the change in y is 18 - 12 = 6, and the change in x is 6 - 3 = 3.

Therefore, the ratio of the change in y to the change in x is 6/3 = 2.

Since y = kx, we know that k = 2.

Now, we can substitute the value of k into the equation to find y when x = 4:

y = kx
y = 2 * 4
y = 8

Voila! When x = 4, y = 8. And no, this isn't a joke!

To find the law connecting the variables, we'll first calculate the rate of change of y with respect to x.

When x changes from 3 to 6, y changes from 12 to 18. The change in y is 18 - 12 = 6, and the change in x is 6 - 3 = 3.

Therefore, the rate of change of y with respect to x is 6/3 = 2.

This tells us that when x increases by 1, y increases by 2. So, the law connecting the variables is y = 2x + c, where c is a constant.

Now, we can substitute the given values (y=12,x=3) into the equation to find the value of c:

12 = 2(3) + c
12 = 6 + c
c = 12 - 6
c = 6

Hence, the law connecting the variables is y = 2x + 6.

To find the value of y when x=4, we can substitute this value into the equation:

y = 2(4) + 6
y = 8 + 6
y = 14

Therefore, when x = 4, y = 14.

To find the law connecting the variables, we need to determine the relationship between y and x based on the given data points.

First, let's compare the values of y when x varies from 3 to 6:

When x = 3, y = 12
When x = 6, y = 18

We can observe that as x doubles from 3 to 6, y also doubles from 12 to 18. This suggests that the relationship between y and x is linear.

Now, let's find the gradient or slope of the linear relationship:

Gradient = (change in y) / (change in x)
= (18 - 12) / (6 - 3)
= 6 / 3
= 2

Since y varies as x with a gradient of 2, we can write the equation as:
y = 2x + C

To determine the value of the constant C, we can substitute one of the given data points into the equation. We'll use the point (x = 3, y = 12):

12 = 2(3) + C
12 = 6 + C

By solving this equation, we find that C = 6.

Therefore, the law connecting the variables is:
y = 2x + 6

Now, to find the value of y when x = 4, we can substitute this value into the equation we just found:

y = 2(4) + 6
y = 8 + 6
y = 14

Hence, when x = 4, y = 14.