Find two consecutive even integers such that three times the lesser integer, added to the larger integer, is 58?

Let's allow "x" to represent the first even integer and "x + 2" to represent the consecutive even integer:

3x + (x + 2) = 58  (your statement)
4x + 2 = 58  (collect terms)
4x + 2 - 2 = 58 - 2  (subtract 2 from both sides)
4x = 56  (collect terms)
4x/4 = 56/4  (divide by 4 on both sides to isolate the variable)
x = 14
x + 2 = 16

The two consecutive even integers you seek are 14 and 16.