Question: Salma, Austin, and Tony sent a total of 129 text messages over their cell phones during the…

Salma, Austin, and Tony sent a total of 129 text messages over their cell phones during the weekend. Austin sent 9 fewer messages than Salma. Tony sent 4 times as many messages as Austin. How many messages did they each send?

Number of text messages Salma sent:

Number of text messages Austin sent:

Number of text messages Tony sent:

Solution

To solve the problem, we will define variables and set up a system of equations based on the information given: Let \( x \) be the number of messages Salma sent. Let \( y \) be the number of messages Austin sent. Let \( z \) be the number of messages Tony sent. From the problem statement, we have these equations: 1. The total number of messages sent by Salma, Austin, and Tony is 129. \[ x + y + z = 129 \] 2. Austin sent 9 fewer messages than Salma. \[ y = x - 9 \] 3. Tony sent 4 times as many messages as Austin. \[ z = 4y \] Now, we solve the system of equations. Substitute equation 2 into equation 1: \[ x + (x - 9) + z = 129 \] Substitute equation 3 into the equation above: \[ x + (x - 9) + 4(x - 9) = 129 \] Simplify and solve for \( x \): \[ x + x - 9 + 4x - 36 = 129 \] \[ 6x - 45 = 129 \] Add 45 to both sides: \[ 6x = 174 \] Divide by 6: \[ x = 29 \] Substitute \( x = 29 \) into equation 2 to find \( y \): \[ y = 29 - 9 = 20 \] Substitute \( y = 20 \) into equation 3 to find \( z \): \[ z = 4 \times 20 = 80 \] Therefore, Salma sent 29 messages, Austin sent 20 messages, and Tony sent 80 messages.