### Instruction: The sum of two numbers is $12$ and their difference is $20$. What is the smaller number?### Response: To solve this problem, I can use a system of linear equations.
Let x be the larger number and y be the smaller number.
Then I have the equations x + y = 12 and x - y = 20.
To eliminate y, I can add the two equations and get 2x = 32.
Dividing both sides by 2, I get x = 16.
To find y, I can substitute x = 16 into either equation.
For example, using x + y = 12, I get 16 + y = 12.
Subtracting 16 from both sides, I get y = -4.
This means the smaller number is -4.