Working together, Joey and his sister, Leslie, can paint a dining room in $1\frac{1}{2}$ days. They can also paint the dining room completely if Joey paints the room for 1 day and Leslie for 3 days. How long will it take each of them to paint the dining room alone?

Let x be the number of days it will take Joey to paint the dining room alone, and y be the number of days it will take Leslie to paint the dining room alone.

Apply the work equation previously discussed. Since Joey and Leslie can finish painting the dining room together for $1\frac{1}{2}$ days, you will have

Moreover, if Joey paints the dining room for 1 day and Leslie for 3 days, the room is completely painted. Hence,

Let $s=\frac{1}{x}$ and $t=\frac{1}{y}.$ Substituting these in equations (1) and (2) will give you the following system of two linear equations in two variables:

You now have a system of two linear equations in variables s and t. To solve for s and t, multiply equation (3) by –1, and then add the resulting equation to equation (4).

Thus, $t=\frac{1}{6}.$ To solve for s, let $t=\frac{1}{6}$ in either equation (3) or (4). Try equation (3), which is $s+t=\frac{2}{3}.$

Hence, $s=\frac{1}{2}.$ Going back to the original problem, solve for x and y by substituting the values of s and t in $s=\frac{1}{x}$ and in $t=\frac{1}{y}.$

Thus, it will take Joey and Leslie 2 days and 6 days, respectively, to paint the dining room working alone. Check the answers as an exercise.