15. If \( K \) is the midpoint of \( \overline{J L}, J K=8 x+11 \) and \( K L=14 x-1 \), find \( J L \).

To solve for \( JL \), we first use the fact that \( K \) is the midpoint of \( \overline{JL} \). This means that the lengths \( JK \) and \( KL \) are equal. Therefore, we set \( JK \) equal to \( KL \): \[ JK = KL \] This translates to: \[ 8x + 11 = 14x - 1 \] Now, we can solve for \( x \). First, we subtract \( 8x \) from both sides: \[ 11 = 6x - 1 \] Next, we add \( 1 \) to both sides: \[ 12 = 6x \] Now, divide both sides by \( 6 \): \[ x = 2 \] Now that we have the value of \( x \), we can find the lengths of \( JK \) and \( KL \) using this value. Start by substituting \( x \) back into the expressions for \( JK \) and \( KL \): \[ JK = 8(2) + 11 = 16 + 11 = 27 \] \[ KL = 14(2) - 1 = 28 - 1 = 27 \] Since \( JK \) equals \( KL \) as expected, we can now find the total length of \( JL \): \[ JL = JK + KL = 27 + 27 = 54 \] Thus, the length of \( JL \) is: \[ \boxed{54} \]