**Solution:**

To find the value of a² + b², we need to first determine the values of a and b.

Given:
a - b = 11 ...(1)
a + b = 13 ...(2)

We can solve this system of equations using the method of substitution or elimination. Let's use the method of substitution.

**Method of Substitution:**

Step 1: Solve equation (1) for a in terms of b:
a = b + 11 ...(3)

Step 2: Substitute the value of a from equation (3) into equation (2):
(b + 11) + b = 13
2b + 11 = 13
2b = 13 - 11
2b = 2
b = 2/2
b = 1

Step 3: Substitute the value of b into equation (3) to find a:
a = 1 + 11
a = 12

Therefore, the values of a and b are a = 12 and b = 1.

**Finding the value of a² + b²:**

Now, we can substitute the values of a and b into the expression a² + b² to find the required value.

a² + b² = 12² + 1²
= 144 + 1
= 145

Therefore, the value of a² + b² is 145.

To summarize:
- Given equations: a - b = 11 and a + b = 13
- Using the method of substitution, we find a = 12 and b = 1
- Substituting these values into the expression a² + b², we get 145 as the answer.

(a - b) ^2=a^2+b^2 - 2ab
11^2=a^2+b^2 - 2×13
a^2+b^2=121+26=147