Fourteen more than twice the first integer gives the second integer. 

          Replace the words "the first integer" by "x".
          Replace the words "the second integer" by "y".

Fourteen more than twice x gives y.

          Replace the words "twice x" by "2x".

Fourteen more than 2x gives y.

          Replace the words "Fourteen more than 2x" by "2x+14". 

2x+14 gives y.

          Replace the word "gives" by an equal sign"

2x+14 = y

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The second integer increased by one is the square of the first integer. 

          Replace the words "the second integer" by "y".
          Replace the words "the first integer" by "x".

y increased by one is the square of x.

          Replace the words "y increased by one" by "y+1".
          Replace the words "the square of x" by "x²".

y+1 is x²

          Replace the word "is" by an equal sign:

y+1 = x²

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          Equations:

2x+14 = y
y+1 = x²

          Solutions of system of equations:

Substitute 2x+14 for y in the second equation:

(2x+14)+1 = x²
  2x+14+1 = x²
  2x+15 = x²
      0 = x²-2x-15
      0 = (x+3)(x-5)

    x+3 = 0;   x-5 = 0
     x = -3;    x = 5

   2x+14 = y      2x+14 = y
2(-3)+14 = y    2(5)+14 = y
   -6+14 = y      10+14 = y
       8 = y         24 = y

          Solutions:
x=-3, y=8   and   x=5, y=24

          Edwin