The sum of two numbers is thirty seven. Using x to represent the larger of two numbers, translate "the difference between nine more than twice the larger number and the sum of the smaller number and three" into a variable expression

Two numbers = 37

Larger number is x, so the smaller number is 37 - x:

x + (37 - x) = 37

"The difference" is subtraction, so we are subtracting the smaller and larger number.

"Nine more than twice the larger number" looks like 9 + 2x, and "the sum of the smaller number and three" looks like (37 - x) + 3. Because this is subtraction, we would write (9 + 2x) - (37 - x) + 3. First, distribute the negative sign:

- (37 - x) = - 37 + x

We then simplify:

9 + 2x - 37 + x + 3 (subtract 3)

9 + 2x - 34 + x

-25 + 2x + x (subtract 8 and 34 then add 2x and x)

-25 + 3x