Let's solve the problem step by step.

Problem Information:

• Last Month:

  • Morning route: $16$ times.
  • Afternoon route: $12$ times.
  • Total distance: $141$ miles.

• This Month:

  • Morning route: $10$ times.
  • Afternoon route: $15$ times.
  • Total distance: $123.75$ miles.

Step 1: Set Variables

Let:

• $x$: Distance of the morning route (in miles).
• $y$: Distance of the afternoon route (in miles).

Step 2: Create Equations

From the problem:

1. $16x + 12y = 141$ (Equation 1 for last month)
2. $10x + 15y = 123.75$ (Equation 2 for this month)

Step 3: Solve the System of Equations

Simplify Both Equations:

1. Divide Equation 1 by $4$: 4x + 3y = 35.25 \tag{3}

2. Divide Equation 2 by $5$: 2x + 3y = 24.75 \tag{4}

Subtract Equation 4 from Equation 3:

$(4x + 3y) - (2x + 3y) = 35.25 - 24.75$ $2x = 10.5 \implies x = 5.25$

Step 4: Verify the Answer

Substitute $x = 5.25$ into Equation 3: $4(5.25) + 3y = 35.25$ $21 + 3y = 35.25 \implies 3y = 14.25 \implies y = 4.75$

Final Answer:

• Morning route ($x$) = 5.25 miles
• Afternoon route ($y$) = 4.75 miles

Would you like further clarification or additional examples?

Here are 5 related questions for practice:

1. How would you solve a similar problem with different frequencies for the routes?
2. What happens if there is only one route available?
3. Can you set up a system of equations for a problem involving three types of routes?
4. How can you confirm that the calculated solution satisfies both equations?
5. How can this approach be applied to distance-rate-time problems?

Tip: Always simplify equations whenever possible to make solving easier!