 In a family, the ages of two siblings, Alex and Jamie, and their mother, Mrs. Smith are described 
by the following conditions:
Alex is 3 years younger than Jamie.
Mrs. Smith is 4 times as old as Alex.
The sum of Alex’s and Jamie’s ages is 27 years.
a) Write a linear equation representing Jamie’s age, if Alex’s age is x years.
b) Write a linear equation representing Mrs. Smith’s age in terms of Alex’s age.
c) Find the ages of Jamie and Mrs. Smith.

Let's break this down step by step.

We are given the following conditions:
- Alex is 3 years younger than Jamie.
- Mrs. Smith is 4 times as old as Alex.
- The sum of Alex's and Jamie's ages is 27 years.

### Part (a): Write a linear equation representing Jamie’s age if Alex’s age is \(x\) years.

Let Alex's age be \(x\) years. From the first condition, Jamie is 3 years older than Alex. Therefore, Jamie’s age is:
\[
\text{Jamie’s age} = x + 3
\]
So, the linear equation representing Jamie’s age is:
\[
\boxed{J = x + 3}
\]

### Part (b): Write a linear equation representing Mrs. Smith’s age in terms of Alex’s age.

From the second condition, Mrs. Smith is 4 times as old as Alex. Therefore, Mrs. Smith's age is:
\[
\text{Mrs. Smith’s age} = 4x
\]
So, the linear equation for Mrs. Smith’s age is:
\[
\boxed{M = 4x}
\]

### Part (c): Find the ages of Jamie and Mrs. Smith.

We are also given that the sum of Alex's and Jamie's ages is 27 years. Therefore, the sum of their ages is:
\[
x + (x + 3) = 27
\]

Simplifying this equation:
\[
2x + 3 = 27
\]
Subtract 3 from both sides:
\[
2x = 24
\]
Divide both sides by 2:
\[
x = 12
\]
So, Alex is 12 years old.

Now, we can find the ages of Jamie and Mrs. Smith:
- Jamie’s age is \( x + 3 = 12 + 3 = 15 \) years.
- Mrs. Smith’s age is \( 4x = 4 \times 12 = 48 \) years.

Thus:
- Alex is 12 years old.
- Jamie is 15 years old.
- Mrs. Smith is 48 years old.

---

Would you like further details or have any questions?

Here are 5 related questions:
1. What would be Jamie’s age if Alex were 10 years old?
2. How does the relationship between the ages of Alex and Jamie affect the solution?
3. How would the equation change if Mrs. Smith were 5 times as old as Alex instead of 4 times?
4. If the sum of Alex's and Jamie's ages were 30 instead of 27, what would their ages be?
5. How can linear equations be used to represent age relationships in other real-life scenarios?

**Tip:** Always organize the given information into equations. It makes solving for unknowns much easier!

In a family, the ages of two siblings, Alex and Jamie, and their mother, Mrs. Smith are described by the following conditions: Alex is 3 years younger than Jamie. Mrs. Smith is 4 times as old as Alex. The sum of Alex’s and Jamie’s ages is 27 years.
a) Write a linear equation representing Jamie’s age, if Alex’s age is x years.
b) Write a linear equation representing Mrs. Smith’s age in terms of Alex’s age.
c) Find the ages of Jamie and Mrs. Smith.

Learn how to solve a family age problem using linear equations. Alex is 3 years younger than Jamie, and Mrs. Smith is 4 times as old as Alex. Discover how to represent these relationships algebraically and find their exact ages. Suitable for students in Grades 6-8.

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