Violet knows that her subscription rate for the basic tier of cable television increased from $18.20 to $19.50 at some point during the calendar year. she paid a total of $230.10 for the year to the cable company. determine the month of the rate

The 2 equations are
18.20x+19.50y=230.10
and
x+y=12
where x is the months of original cost and y is months for new cost. Since you know that you paid for one year (12 months) you can make the second equation. Then you want to substitute the first equations x by making the second equation
x=(12-y)
18.20(12-y)+19.50y=230.10
218.40-18.20y+19.50y=230.10
1.30y=11.70
y=9
so that means you had the original rate for 3 months and the new one for 9 months