Let's break down the information given in the problem:

• $LK = MK$
• $LK = 7x - 10$
• $KN = x + 3$
• $MN = 9x - 11$
• $KJ = 28$

The goal is to find the length of $LJ$.

Step 1: Use the fact that $LK = MK$

Since $LK = MK$, we know that:

$LK + KN = MN$

Substitute the expressions for $LK$, $KN$, and $MN$:

$(7x - 10) + (x + 3) = 9x - 11$

Step 2: Simplify the equation

Combine like terms on the left-hand side:

$8x - 7 = 9x - 11$

Now, solve for $x$:

Step 3: Solve for $x$

Subtract $8x$ from both sides:

$-7 = x - 11$

Add 11 to both sides:

$4 = x$

Step 4: Substitute $x = 4$ back into the expressions

Now, substitute $x = 4$ into the equations for $LK$, $KN$, and $MN$:

• $LK = 7(4) - 10 = 28 - 10 = 18$
• $KN = 4 + 3 = 7$
• $MN = 9(4) - 11 = 36 - 11 = 25$

Step 5: Find the total length of $LJ$

To find $LJ$, we need to sum the lengths of $LK$, $KJ$, and $KN$:

$LJ = LK + KJ + KN = 18 + 28 + 7 = 53$

Thus, the length of $LJ$ is:

$\boxed{53}$

Would you like more details on any part of this solution or have any questions?

Here are some related questions to expand on the topic:

1. What if $KJ$ was not given? Could we find it using other information?
2. How would the problem change if $LK \neq MK$?
3. Can we find $MJ$ given that $MN = 25$?
4. What is the importance of knowing that $LK = MK$?
5. What types of geometric figures might this represent?

Tip: Always remember to check for consistency in given conditions when solving equations involving segments.