This is a system of equations. There are at least 4 methods available to solve this system: graphing, substitution, linear combinations (also called elimination or the addition method), or using a matrix equation. Since there are fractions, I would recommend linear combinations over the first two methods and the fourth requires a graphing calculator, so here goes:
x + 2y = 2/3
2x + y = 3/2
First you must decide whether you want to eliminate the x's or the y's. Either is a good choice since 2 times x = 2x or 2 times y = 2y. We want to create a pair of opposites. I am going to choose to eliminate the y's.
Multiply the second equation by -2.
-2(2x + y) = -2(3/2), which simplifes to -4x - 2y = -3. Now add this equation to the first equation (x + 2y = 2/3).
-4x + x + (- 2y) + 2y = -3 + 2/3, simplifies to -3x = -7/3 since -3 = -9/3 and you need a common denominator to add fractions.
Now divide both sides by -3 and you get x = 7/9 - we're half way there, now we need to use substitution to find y.
Replace x in either equation with 7/9 and solve for y. I am going to use the first equation...
7/9 + 2y = 2/3, but 2/3 = 6/9 so 7/9 + 2y = 6/9, subtract 7/9 from both sides to isolate the 2y
2y = -1/9, now divide both sides by 2 (or multiply both sides by 1/2) and we get y = -1/18.
That makes the solution (7/9, -1/18). Be sure to check your solution in both of the original equations.
Check: 7/9 + 2(-1/18) = 7/9 - 1/9 = 6/9 = 2/3
Check: 2(7/9) + (-1/18) = 14/9 - 1/18 = 28/18 - 1/18 = 27/18 which reduces to 3/2.
The solution checks in both equations.