The perimeter of square a is 28 feet less than the perimeter of square b. the area of square a is 161 square feet less than the area of square b. if x and y are the side lengths of square a and square b, respectively, what is the value of x + y?

X+y=23.

The equation comparing the areas of the squares would be:

x²=y²-161

This is because the area of a square is found by multiplying its length and width; if x is the length of the smaller square, and y is the length of the larger square, we know that x*x, the area of the smaller square, is 161 less than y*y, the area of the larger square.

The equation comparing the perimeters of the squares would be:

4x=4y-28

This is because the perimeter is found by adding all of the sides; x+x+x+x=4x.  Similarly, y+y+y+y=4y; however, the perimeter of the smaller square is 28 less than 4y.

Using the equation for the perimeters, we can isolate x.  Divide both sides by 4:

4x/4 = 4y/4 - 28/4
x=y-7

Now we can substitute this into the equation for area:
(y-7)²=y²-161
(y-7)(y-7)=y²-161

Multiplying the binomials on the left we have:
y*y-7*y-7*y-7(-7)=y²-161
y²-7y-7y+49=y²-161
y²-14y+49=y²-161

We cannot have a variable on both sides; subtract y² from each:
y²-14y+49-y²=y²-161-y²
-14y+49= -161

Subtract 49 from both sides:
-14y+49-49 = -161-49
-14y = -210

Divide both sides by -14:
-14y/-14 = -210/-14
y=15

Substitute this back into the perimeter equation:
4x=4(15)-28
4x=60-28
4x=32

Divide both sides by 4:
4x/4=32/4
x=8

Therefore x+y=8+15=23.