What is the distance of the morning route in miles?

Steps to Solve

1. Define Variables Let ( x ) represent the distance of the morning route in miles and ( y ) represent the distance of the afternoon route in miles.

2. Set Up Equations for Last Month From the information provided:

• The mail carrier delivered on the morning route 16 times and on the afternoon route 12 times, totaling 141 miles. This gives us the equation: $$ 16x + 12y = 141 $$

3. Set Up Equations for This Month For this month, the mail carrier delivered on the morning route 10 times and on the afternoon route 15 times, totaling 123.75 miles. This results in the equation: $$ 10x + 15y = 123.75 $$

4. Simplify the Second Equation We can simplify the second equation by dividing all terms by 5: $$ 2x + 3y = 24.75 $$

5. Solve the System of Equations Now we solve the system of equations:

1. ( 16x + 12y = 141 )
2. ( 2x + 3y = 24.75 )

We can solve the second equation for ( y ): $$ 3y = 24.75 - 2x $$ $$ y = \frac{24.75 - 2x}{3} $$

6. Substitute ( y ) in the First Equation Now substitute the expression for ( y ) into the first equation: $$ 16x + 12 \left( \frac{24.75 - 2x}{3} \right) = 141 $$

7. Distribute and Solve for ( x ) Multiply through: $$ 16x + 4(24.75 - 2x) = 141 $$ This simplifies to: $$ 16x + 99 - 8x = 141 $$ Combining like terms: $$ 8x + 99 = 141 $$

8. Isolate ( x ) Now isolate ( x ): $$ 8x = 141 - 99 $$ $$ 8x = 42 $$ $$ x = \frac{42}{8} = 5.25 $$

9. Determine ( y ) if Needed Substituting ( x ) back into either equation can yield ( y ), but it isn’t necessary for the answer.