https://file.aiquickdraw.com/mathbot/file/1726753616577mepoym6d.jpg

Let's solve the problems in the image step by step:

### Problem 1: Shop Discount
- The shop discounts all products by $10. The original price of a coat is $ c $, and the sale price is $59.
  
We are given:
\[
59 = c - 10
\]

To find $ c $ (the original price), solve for $ c $:
\[
c = 59 + 10 = 69
\]
Thus, the original price of the coat is **$69**.

### Problem 2: Taxi Fare
- The taxi fare has a fixed charge of $2.70, and the fare increases by $3 per mile. A journey of $ m $ miles costs $32.70.

We know:
\[
32.70 = 2.70 + 3m
\]

To find $ m $, subtract 2.70 from both sides:
\[
32.70 - 2.70 = 3m
\]
\[
30 = 3m
\]

Now, divide both sides by 3:
\[
m = \frac{30}{3} = 10
\]
Thus, the journey was **10 miles** long.

### Problem 3: Ages of Luke, Kate, and Adam
- Luke is $ x $ years old, Kate is half of Luke's age, and Adam is 5 years younger than Kate. Adam is 5 years old.

Let’s break it down:
- Kate's age is $ \frac{x}{2} $,
- Adam’s age is $ \frac{x}{2} - 5 $,
- Adam is given as 5 years old.

We can set up the equation:
\[
\frac{x}{2} - 5 = 5
\]

Solve for $ x $:
\[
\frac{x}{2} = 5 + 5 = 10
\]
Multiply both sides by 2:
\[
x = 10 \times 2 = 20
\]

Thus, **Luke is 20 years old**, and from this:
- Kate is $ \frac{20}{2} = 10 $ years old.
- Adam is $ 10 - 5 = 5 $ years old, as given.

---

Would you like further clarification on any of these solutions?

Here are 5 related questions to extend your understanding:
1. What would the sale price of the coat be if the discount were $15 instead of $10?
2. If the taxi rate increased by $1 per mile, what would the new total cost of a 10-mile journey be?
3. If Kate was 2/3 of Luke's age instead of half, how old would Luke be?
4. If the cost of a coat was reduced by 20%, what would the original price need to be to match the same $59 sale price?
5. How many miles would the taxi ride be if the cost was $50 under the same fare conditions?

**Tip:** When solving linear equations, isolate the variable by performing the inverse operations step by step.

Solve the following problems:
1. A shop discounts all products by $10 in a sale. The original price of a coat is $c, and the sale price is $59.
2. The cost of a taxi fare is a fixed charge of $2.70 plus $3 per mile. A journey of m miles costs $32.70.
3. Luke is x years old. Kate is half Luke’s age. Adam is 5 years younger than Kate. Adam is 5 years old. Find the age of Luke.

Learn how to solve algebraic word problems involving shop discounts, calculating taxi fares, and determining ages with equations. This step-by-step guide is suitable for students in Grades 6-8.

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