A linear sequence starts: a+2b, a+6b, a+10b, ..., ... The 2nd term has a value of 8. The 5th term has a value of 44. Work out the values of a and b.

We know the values for the 2nd and 5th terms and already have the 2nd term as an equation but not the 5th. Therefore, we need to start by working out what exactly the 5th term is in terms of a and b. From looking at terms 1-3 we can tell they increase by 4b every time. So, the 5th term must be:
(a+10b) + 4b + 4b = a + 18b
Next let's set up our simultaneous equations for the 2nd and 5th terms now we have all the information.
(1) a + 6b = 8 (2) a + 18b = 44
I have labelled the equations (1) and 2. We need to eliminate either the a or b from each equation. This is most easily done by subtracting equation (1) from equation (2).
(1) - (2) = 12b = 36 b = 3
Now that we have the value for b, let's substitute it into equation (1).
a + (6x3) = 8a + 18 = 8 a = -10 Answer: a = -10 b = 3