To solve this problem, let's break down each scenario and set up equations based on the information provided.

Problem Summary

1. First week:

   • Henry drives to work and back 5 times.
   • He drives to the supermarket and back 2 times.
   • The total distance he drives is 72 km.

2. Second week:

   • Henry drives to work and back 4 times.
   • He drives to the supermarket and back 3 times.
   • The total distance he drives is 66 km.

Let:

• $w$ = the round-trip distance to work (in km).
• $s$ = the round-trip distance to the supermarket (in km).

Setting up the Equations

From the information given, we can set up the following equations:

1. For the first week: $5w + 2s = 72$

2. For the second week: $4w + 3s = 66$

Solving the System of Equations

We can solve these equations simultaneously to find the values of $w$ and $s$.

Step 1: Multiply the equations to align the terms for elimination

To make the $w$ terms cancel out, we can multiply the first equation by 3 and the second equation by 2:

$3(5w + 2s) = 3 \times 72$ $2(4w + 3s) = 2 \times 66$

This results in:

1. $15w + 6s = 216$
2. $8w + 6s = 132$

Step 2: Subtract the equations

Now subtract the second equation from the first to eliminate $s$:

$(15w + 6s) - (8w + 6s) = 216 - 132$ $7w = 84$ $w = 12$

Step 3: Substitute $w = 12$ back into one of the original equations

Using the first equation: $5w + 2s = 72$ $5(12) + 2s = 72$ $60 + 2s = 72$ $2s = 12$ $s = 6$

Solution

• The round-trip distance to work, $w$, is 12 km.
• The round-trip distance to the supermarket, $s$, is 6 km.

Final Answers

a) The distance to and from work is 12 km.

b) The distance to and from the supermarket is 6 km.

Would you like further details on this solution, or do you have any questions?

Related Questions

1. How would the equations change if the distances in the second week were different?
2. What if Henry drove to work more times in one of the weeks—how would that affect the total?
3. How can you set up similar equations for different types of transportation?
4. How can we verify the solution by plugging values back into the equations?
5. What is a general approach for solving simultaneous linear equations with two variables?

Tip

When solving word problems, clearly define your variables and set up equations based on the information provided. This helps simplify the problem systematically.