Write an efficient, not brute force, program that solves the following problem: suppose we have two unknown quantities x and y. If three time x plus two times y is twenty, and 2 times x plus three time y is ten, what are x and y?
Output:
One way to solve this problem efficiently is to use the method of elimination. We can set up a system of equations as follows:
3x + 2y = 20
2x + 3y = 10
We can then multiply the first equation by 2 and the second equation by 3, to get:
6x + 4y = 40
6x + 9y = 30
We can then subtract the second equation from the first equation, to get:
-5y = 10
y = -2
We can then substitute this value of y into one of the original equations (for example, the first one), to get:
3x + 2(-2) = 20
3x - 4 = 20
3x = 24
x = 8
Therefore, the values of x and y are x = 8 and y = -2.
I tried your prompt 5 times in a row using the Try Again button. The results:
x = 10 and y = 5 (adding equations together)
x = 10/33 and y = 15/11 (substitution method)
x = 5.33 and y = 2 (substitution method)
x =5 and y = 2.5 (algebraic manipulation)
x = 5 and y = 5 (elimination method)
However, when I asked it to write a python program to solve the problem, it did much better:
Write a Python program using numpy to solve the following problem: suppose
we have two unknown quantities x and y. If three time x plus two times y is
twenty, and 2 times x plus three time y is ten, what are x and y?
It produced the following program:
import numpy as np
# Define the matrix of coefficients
A = np.array([[3, 2], [2, 3]])
# Define the vector of constants
b = np.array([20, 10])
# Solve the system of equations
x = np.linalg.solve(A, b)
print(x)
Which is basically correct. (The only nitpick I can see is that `linalg.solve` will return a vector containing both x and y, so a better answer would be `x, y = np.linalg.solve(A, b)`.) If you copy-paste the above program you do in fact get "[8. -2.]", which is correct.
However, ChatGPT, after providing the correct program, also claimed that it's output would be "[5. 5.]" which is not correct.
My impression is that ChatGPT being a large language model, is excellent at translating from English to Python, but terrible at actually performing calculations. Which is fine. We have programs which can efficiently run numerical programs. ChatGPT fills the role of a programmer, not a calculator.
I want to emphasize how impressive I think ChatGPT is. Even the above examples, where it gets the "wrong" answer in the end, are impressive. Most of my interactions with it were very positive. But we need to understand its strengths and weaknesses to be able to use it effectively.
It does go off the rails sometimes, but this is kind of correct. Prompt:
Write an efficient, not brute force, python program that solves the following problem: suppose we have two unknown quantities x and y. If three time x plus two times y is twenty, and 2 times x plus three time y is ten, what are x and y?
Output:
To solve this problem efficiently, we can use linear algebra. We can represent the equations as a system of linear equations, and then use matrix operations to solve for the unknown quantities x and y.
First, we can represent the equations as follows:
3x + 2y = 20
2x + 3y = 10
Next, we can put the equations in matrix form:
| 3 2 | | x | | 20 |
| 2 3 | * | y | = | 10 |
To solve for x and y, we can use the inverse of the matrix on the left side to eliminate the coefficients. The inverse of a matrix is denoted as A^-1, and is calculated as follows:
A^-1 = (1/det(A)) * adj(A)
Where det(A) is the determinant of the matrix A, and adj(A) is the adjugate matrix of A. The determinant of a 2x2 matrix is calculated as follows:
det(A) = a * d - b * c
Where a, b, c, and d are the elements of the matrix A:
| a b |
| c d |
The adjugate matrix of a 2x2 matrix is calculated as follows:
adj(A) = | d -b |
| -c a |
We can then calculate the inverse of the matrix on the left side of the equations, and use it to solve for x and y:
| 3 2 |^-1 1 | 3 -2 |
| 2 3 | ----- * | -2 3 | = (1/5) * | 3 -2 | = | 3/5 -2/5 |
|-2 3 | |-2/5 3/5 |
| 3/5 -2/5 | | x | | 20 |
|-2/5 3/5 | * | y | = | 10 |
We can then solve for x and y by multiplying the inverse matrix with the vector on the right side of the equations:
| 3/5 -2/5 | | 20 |
|-2/5 3/5 | * | 10 | = | 6 |
| 7 |
Therefore, the values of x and y are 6 and 7, respectively.
Here is the complete Python program to solve this problem using linear algebra:
Calculate the determinant of a 2x2 matrix
def determinant(a, b, c, d):
return a * d - b * c
Calculate the adjugate matrix of a 2x2 matrix
def adjugate(a, b, c, d):
return [[d, -b], [-c, a]]
Calculate the inverse of a 2x2 matrix
def inverse(a, b, c, d):
det = determinant(a, b, c, d)
adj = adjugate(a, b, c, d)
return [[adj[0][0] / det, adj[0][1] / det], [adj[1][0] / det, adj[1][1] / det]]
Solve the system of linear equations
def solve(a, b, c, d, e, f):
inv = inverse(a, b, c, d)
return [inv[0][0] * e
Mr. Brian Simms, K.C. (Lennox Paton Counsel and Attorney-at Law) was appointed as provisional liquidator. Additionally, the powers of the directors of FDM have been suspended and no assets of FDM, client assets or trust assets held by FDM, can be transferred, assigned or otherwise dealt with, without the written approval of the provisional liquidator.
You mean predictions that can be tested right now? Unclear why the proposed time horizon is the right one. This standard would have rejected most of quantum mechanics, some of GR, and many other useful theories and explanations when they were first proposed.
Correct. Similarly, one can imagine a point in the future in which technology advances enough to prove/disprove various predictions with regard to psychology.
You and the person you are responding to are in agreement. They were being sarcastic.
Science definitely involves explanation as well as prediction, otherwise we’d still be iterating on more and more complex epicycles as predictive models for astrophysics. Induction isn’t everything.
