Question Analysis:
We are given 150 weights, some weighing 1 kg and some weighing 2 kg. The sum of the weights is 260 kg. We need to determine the number of 1 kg weights.

Given:
- Total number of weights = 150
- Sum of all weights = 260 kg

Solution:
Let's assume the number of 1 kg weights as 'x' and the number of 2 kg weights as 'y'.

We know that the sum of the weights is 260 kg. Therefore, we can write the equation:

1*x + 2*y = 260

We also know that the total number of weights is 150. So, we can write another equation:

x + y = 150

Now, we have a system of two equations with two variables. We can solve this system to find the values of 'x' and 'y'.

Solving the Equations:

Equation 1: x + 2y = 260
Equation 2: x + y = 150

We can solve this system of equations using the method of substitution or elimination.

Method 1: Substitution
We can solve Equation 2 for x and substitute it in Equation 1.

From Equation 2, we have: x = 150 - y

Substituting this value of x in Equation 1, we get:

(150 - y) + 2y = 260
150 + y = 260
y = 260 - 150
y = 110

Now, substitute the value of y back into Equation 2 to find x:

x + 110 = 150
x = 150 - 110
x = 40

Method 2: Elimination
We can eliminate one variable by subtracting Equation 2 from Equation 1.

Equation 1: x + 2y = 260
Equation 2: -x - y = -150

Adding these two equations, we get:

x + 2y - x - y = 260 - 150
y = 110

Substitute the value of y back into Equation 2 to find x:

x + 110 = 150
x = 150 - 110
x = 40

Conclusion:
After solving the system of equations, we find that there are 40 weights weighing 1 kg.