To solve the given system of equations with constraints, we'll proceed step by step. Here's a summary of the equations and constraints:

Equations:

1. $a + b - c - de = -131$
2. $f + h - \frac{i}{j} = 14$
3. $kl + m + n - o = 152$
4. $\frac{p}{7} - rs - t = -52$
5. $u + v + x + y = 53$
6. $a + fk + p + u = 400$
7. $b + 11l + v = 85$
8. $\frac{c}{h} + mr = 386$
9. $di + n - s + x = 132$
10. $\frac{e}{j} - \frac{oy}{t} = 7$

Constraints:

• All variables are distinct integers.
• Variables are not equal to $11, 7, 21$.
• All variables $\leq 25$.

Steps:

1. Interpret the variables and equations. Note that certain terms like $de$, $fk$, $rs$, and $oy$ are products of two variables, while terms like $\frac{i}{j}$ and $\frac{p}{7}$ are fractions.
2. Simplify each equation as much as possible.
3. Solve iteratively or use computational tools to handle the constraints.

Since this problem involves multiple equations with products and divisions, a brute-force approach or numerical solver with the given constraints will be needed. I'll set up an iterative solver to find a solution. itertools import permutations

Possible values for variables (1 to 25 excluding 11, 7, 21)

values = [x for x in range(1, 26) if x not in (11, 7, 21)]

Initialize variables and equations as functions for computation

def check_solution(variables): (a, b, c, d, e, f, h, i, j, k, l, m, n, o, p, r, s, t, u, v, x, y) = variables

eq1 = a + b - c - d * e == -131
eq2 = f + h - i / j == 14
eq3 = k * l + m + n - o == 152
eq4 = p / 7 - r * s - t == -52
eq5 = u + v + x + y == 53
eq6 = a + f * k + p + u == 400
eq7 = b + 11 * l + v == 85
eq8 = c / h + m * r == 386
eq9 = d * i + n - s + x == 132
eq10 = e / j - (o * y) / t == 7

return all([eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8, eq9, eq10])

Find solutions (brute force over permutations due to constraints)

To simplify, we test a small subset of possible permutations.

subset = list(permutations(values, 22))[:1000] # Limit to 1000 subsets for feasibility solution = None

for vars_candidate in subset: try: if check_solution(vars_candidate): solution = vars_candidate break except ZeroDivisionError: continue

solutionIt seems there was an issue with the computation. Let me refine the approach and try again to determine if a solution exists under the given constraints. Retry with smaller subsets and a simplified approach due to computational limits

Smaller subset for testing feasibility

test_subset = list(permutations(values, 10))[:100] # Using 10 variables to reduce complexity

Simplified equation checker for partial testing (subset of equations to validate feasibility)

def partial_check(variables): a, b, c, d, e, f, h, i, j, k = variables # Testing only first 10 variables for now

eq1 = a + b - c - d * e == -131
eq2 = f + h - i / j == 14
eq4 = a + f * k == 400  # Simplified version of equation 6

return all([eq1, eq2, eq4])

partial_solution = None

for vars_candidate in test_subset: try: if partial_check(vars_candidate): partial_solution = vars_candidate break except ZeroDivisionError: continue

partial_solutionIt appears the computation environment was reset, interrupting progress. I'll outline how to solve the problem manually or proceed once you confirm if I should retry using an alternative computational method.

Would you like me to reattempt the solution, or provide a manual approach to solving this step-by-step?