If QR = 11x, RS = 16, and QS = 19x − 8, what is QR?

Step-by-step solution

1. 1

   Apply the segment addition postulate. Since $R$ is between $Q$ and $S$:

   $QR + RS = QS$

   The outer segment $QS$ is the total, so the two inner lengths must add up to it.

2. 2

   Substitute the given expressions.

   $11x + 16 = 19x - 8$

   Both sides contain a multiple of $x$, so the $x$ terms have to be collected before solving.

3. 3

   Collect x on one side and constants on the other.

   $16 + 8 = 19x - 11x \ \Longrightarrow \ 24 = 8x$

   Moving $11x$ to the right keeps the coefficient positive, which avoids a sign error.

4. 4

   Solve for x.

   $x = \frac{24}{8} = 3$

5. 5

   Substitute back to get the requested length.

   $QR = 11x = 11(3) = 33$

   The question asks for $QR$, not $x$, so this substitution is the actual answer.

6. 6

   Check the three lengths. With $x = 3$: $QR = 33$, $RS = 16$, $QS = 19(3) - 8 = 49$, and $33 + 16 = 49$. Consistent and all positive, so $QR = 33$.