Point S is on line segment \( \overline{R T} \). Given \( R S=4 x-10, S T=2 x-10 \), and \( R T=4 x-4 \), determine the numerical length of \( \overline{R S} \). Answer Attempt 1 out of 3 \( R S=\square \) Submit Answer

To solve for the length of \( \overline{R S} \), we can use the information given about the lengths of the segments. The relationship between the segments can be expressed as follows: \[ R S + S T = R T \] Substituting the expressions provided: \[ (4x - 10) + (2x - 10) = (4x - 4) \] Combining like terms on the left side yields: \[ 6x - 20 = 4x - 4 \] Next, we can isolate \( x \) by first subtracting \( 4x \) from both sides: \[ 6x - 4x - 20 = -4 \] \[ 2x - 20 = -4 \] Adding 20 to both sides gives us: \[ 2x = 16 \] Dividing by 2: \[ x = 8 \] Now, we can find \( R S \) by substituting \( x \) back into its expression: \[ R S = 4x - 10 = 4(8) - 10 = 32 - 10 = 22 \] Thus, the numerical length of \( \overline{R S} \) is \[ \boxed{22} \]