To solve for $b$, we will use the given equations systematically. Here's the given data:

1. $a = 13968$
2. $b + c = 7916$
3. $d + e = 6836$
4. $e + f = 10872$
5. $f + g = 15808$
6. $g + h = 16487$
7. $h + i = 22924$
8. $c + i = 19123$
9. $a + h = 21988$
10. $a + b + c = 21884$

Step 1: Calculate $b + c$ and $a + b + c$

From equation $a = 13968$ and $a + b + c = 21884$:

$b + c = 21884 - 13968$ $b + c = 7916$

This matches equation (2), so it is consistent.

Step 2: Solve for $b$

From equation (2):

$b + c = 7916$

Let $c = x$. Then:

$b = 7916 - x$

Step 3: Use $c + i = 19123$ to solve for $i$

From equation (8):

$c + i = 19123$

Substitute $c = x$:

$x + i = 19123 \implies i = 19123 - x$

Step 4: Use $h + i = 22924$ to solve for $h$

From equation (7):

$h + i = 22924$

Substitute $i = 19123 - x$:

$h + (19123 - x) = 22924 \implies h = 22924 - 19123 + x$ $h = 3801 + x$

Step 5: Use $a + h = 21988$ to find $h$

From equation (9):

$a + h = 21988$

Substitute $a = 13968$ and $h = 3801 + x$:

$13968 + 3801 + x = 21988$

Simplify:

$17769 + x = 21988 \implies x = 21988 - 17769$ $x = 4219$

So, $c = 4219$.

Step 6: Solve for $b$

Now substitute $c = 4219$ into $b + c = 7916$:

$b + 4219 = 7916 \implies b = 7916 - 4219$ $b = 3697$

Final Answer:

$b = 3697$

Would you like further explanation or calculations on this problem?

Five Related Questions:

1. What is the value of $c, d, e, f, g, h, i$ based on the other equations?
2. How can we verify the consistency of all the equations given?
3. Can we write the problem as a system of equations and solve using matrix methods?
4. How does this approach generalize to similar problems with more variables?
5. What are other efficient methods to solve simultaneous equations like these?

Tip:

Always start with equations involving the smallest number of variables to simplify the problem systematically