One year ago liz was three times as old as her brother jack. in two years she'll be only twice as old as jack. how old are liz and jack now?

One year ago means you have to take both Liz's age and Jack's age and subtract 1.  In 2 years means you have to take both of their ages and add 2.  L is Liz's age, J is Jack's age.  It tells us that one year ago Liz was 3 times Jack's age, or L = 3J.  But since this was a year ago, it's actually L - 1 = 3(J - 1).  Don't forget that if Liz was a year younger a year ago, so was Jack.  Thats the first equation.  Now it tells us that in 2 years Liz will be twice as old as Jack, or that L = 2J.  But in 2 years they will both be 2 years older.  Hence, L + 2 = 2(J + 2). Distribute through the parenthesis in both equations to get L-1=3J-3, and L+2=2J+4.  Solve the first for L: L=3J-2 and sub that in to the other equation to solve for J: 3J-2+2=2J+4.  Solving for J you get J=4. Now sub that in to solve for L: L = 3(4)-2 and L=10.  Last year they were 3 and 9 (Liz was 3 times older) and in 2 years, Jack will be 6 and Liz will be 12 (Liz will be twice as old as Jack).