Y varies partly as X and as X^2 when X=2 Y=14 and when X=5 Y=65.Express Y interm of X hence find Y when X=-2
How to solve the question
To express Y in terms of X, we'll need to find the relationship between Y and X. Let's start with the given information:
When X = 2, Y = 14.
When X = 5, Y = 65.
From these data points, we can write two equations:
Equation 1: Y = aX + b (Since Y partly varies as X)
Equation 2: Y = cX^2 + d (Since Y partly varies as X^2)
To solve for the coefficients (a, b, c, d), we can substitute the X and Y values from the given data points into these equations.
Substituting X = 2 and Y = 14 in Equation 1:
14 = 2a + b ------- (1)
Substituting X = 2 and Y = 14 in Equation 2:
14 = 4c + d ------- (2)
Substituting X = 5 and Y = 65 in Equation 1:
65 = 5a + b ------- (3)
Substituting X = 5 and Y = 65 in Equation 2:
65 = 25c + d ------- (4)
Now, we have four equations (1)-(4) with four unknowns (a, b, c, d). We can solve this system of equations simultaneously to find the values of a, b, c, and d.
Solving equations (1) and (3), we can eliminate b:
14 = 2a + b (1)
65 = 5a + b (3)
Subtracting equation (1) from equation (3):
65 - 14 = (5a + b) - (2a + b)
51 = 3a
Dividing both sides by 3:
51/3 = a
a = 17
Substituting the value of a back into equation (1):
14 = 2(17) + b
14 = 34 + b
Subtracting 34 from both sides:
-20 = b
So, we have found the values of a and b: a = 17 and b = -20.
To find the values of c and d, we need to solve equations (2) and (4):
Substituting the values of X = 2 and Y = 14 into equation (2):
14 = 4c + d (2)
Substituting the values of X = 5 and Y = 65 into equation (4):
65 = 25c + d (4)
Subtracting equation (2) from equation (4), we can eliminate d:
65 - 14 = (25c + d) - (4c + d)
51 = 21c
Dividing both sides by 21:
51/21 = c
c = 17/7
Substituting the value of c back into equation (2):
14 = 4(17/7) + d
14 = 68/7 + d
Multiplying both sides by 7 to eliminate the fraction:
98 = 68 + 7d
Subtracting 68 from both sides:
30 = 7d
Dividing both sides by 7:
30/7 = d
d = 30/7
We have found the values of c and d: c = 17/7 and d = 30/7.
Now, let's express Y in terms of X based on the determined coefficients:
Y = aX + b
Y = 17X - 20
To find Y when X = -2, we substitute X = -2 into the equation:
Y = 17X - 20
Y = 17(-2) - 20
Y = -34 - 20
Y = -54
Therefore, when X = -2, Y = -54.
y = ax+bx^2
2a+4b = 14
5a+25b = 65
y = 3x+2x^2
take it.