Y is partly constant and partly varies as x. When x=2,y=16 and when x=7,y=31,find y in terms of x.
Y is partly constant and partly varies as x.
So, y = mx+b
Using the points,
2m+b = 16
7m+b = 31
subtract, and you get
5m = 15
m = 3
so, b = 10
y = 3x+10
Well, let's break it down. Y is partly constant and partly varies as x. In other words, we can express y as the sum of a constant term and a term that changes with x. Let's call the constant term "C" and the term that varies with x "k". So the equation for y would be y = C + kx.
Now, we're given two points where x and y values are known: (2, 16) and (7, 31). We can use these points to form two equations to solve for C and k.
From the first point, we have 16 = C + 2k. And from the second point, we have 31 = C + 7k.
Now we have a system of two equations that we can solve simultaneously. Let's solve for C first:
From the first equation, C = 16 - 2k. We'll substitute this value of C into the second equation:
31 = (16 - 2k) + 7k.
Simplifying, we get 31 = 16 + 5k. Subtracting 16 from both sides, we get 15 = 5k. Dividing by 5, we find that k = 3.
Now that we know the value of k, we can substitute it back into the first equation to find C:
16 = C + 2(3).
Simplifying, we get 16 = C + 6. Subtracting 6 from both sides, we find that C = 10.
So, the equation for y in terms of x is y = 10 + 3x. Voila!
To find y in terms of x when y is partly constant and partly varies, we need to determine the constant part and the varying part.
Given that when x = 2, y = 16 and when x = 7, y = 31, we can find the constant part by finding the difference in y for the two values of x.
The difference in x is 7 - 2 = 5, and the difference in y is 31 - 16 = 15.
So, the varying part of y with respect to x is 15/5 = 3.
To find the constant part, we can substitute one of the given values of x and y into the equation.
Using x = 2 and y = 16, we have:
y = constant part + varying part * x
16 = constant part + 3 * 2
16 = constant part + 6
Simplifying, we find:
constant part = 16 - 6 = 10
Now we have the constant part as 10 and the varying part as 3.
Therefore, y in terms of x is:
y = 10 + 3 * x
To find the relationship between y and x, we first need to determine the constant part and the varying part of y.
Let's assume the constant part of y is c, and the varying part is kx, where k is a constant.
Based on the given information, when x = 2, y = 16. Substituting these values into our equation, we get:
16 = c + 2k
Similarly, when x = 7, y = 31:
31 = c + 7k
Now we have a system of equations:
c + 2k = 16
c + 7k = 31
By solving these equations simultaneously, we can find the values of c and k. Subtracting the first equation from the second yields:
c + 7k - (c + 2k) = 31 - 16
5k = 15
k = 3
Substituting this value of k back into either of the equations, we get:
c + 2(3) = 16
c + 6 = 16
c = 10
Now we have found the values of c and k. Therefore, the relationship between y and x is:
y = c + kx
y = 10 + 3x
Therefore, in terms of x, y = 10 + 3x.