This is a classic chicken and rabbit problem. We have 12 ping-pong tables with 34 people total. Some tables have singles matches with 2 people, while others have doubles matches with 4 people. We need to find how many tables of each type there are.
To solve this problem, we need to set up variables and equations. Let x be the number of singles tables and y be the number of doubles tables. From the problem, we know that the total number of tables is 12, so x plus y equals 12. We also know that singles tables have 2 people each and doubles tables have 4 people each, with a total of 34 people, so 2x plus 4y equals 34.
Now let's solve this system of equations using substitution. From the first equation x plus y equals 12, we can solve for x to get x equals 12 minus y. Next, we substitute this expression into the second equation. We get 2 times 12 minus y plus 4y equals 34. Expanding this gives us 24 minus 2y plus 4y equals 34. Simplifying, we get 24 plus 2y equals 34. Subtracting 24 from both sides gives us 2y equals 10, so y equals 5.
Now we can find x by substituting y equals 5 back into our first equation. We get x equals 12 minus 5, which gives us x equals 7. So our solution is 7 singles tables and 5 doubles tables. Let's verify this answer. For the first equation: 7 plus 5 equals 12, which is correct. For the second equation: 2 times 7 plus 4 times 5 equals 14 plus 20, which equals 34. Both equations check out!