This word problem completely lost me. I do not know how to write out the equation and solve the problem... any help?? Thanks!

The sum of three numbers is 10. Twice the smallest is 3 less than the largest, and the sum of the largest and smallest is 6. Use a linear system in three variables to find the three numbers.

I don't get why requiring the use of a "linear system in three variables" is necessary. The equations ARE linear, there are three of them, and there are only three variables.

The sum of three numbers is 10 = a + b + c.

Let a be smallest and c largest. Twice smallest is 2a. Three less than largest is c - 3. So the second equation is 2a = c - 3 or 3 = -2a + c.

The sum of the largest and smallest is 6 = a + c.

The general solution of 3 equations of 3 unknowns is to express one variable in terms of the other two variables narrowing it down to 2 equations in 2 unknowns and then substituting again to get a solution for the one remaining variable. Then you go back and solve for the other variables. In this case a = 6 - c, so 2*(6 - c) = c - 3. c must be 5 and a is 1. That leaves b = 10 - c - a or b = 4.