https://file.aiquickdraw.com/mathbot/file/1727315870655dexti0q3.jpg

Let's work through the problem in the image step by step.

### Problem: 
We are given that \( x \) is an even integer and \( y \) is an odd integer. We are asked to prove that \( 3x + 2y \) is an even integer.

### Solution Outline:

1. **Step 1: Express \( x \) as an even number.**
   Since \( x \) is even, we know that it can be written as \( x = 2j \), where \( j \) is some integer.

2. **Step 2: Express \( y \) as an odd number.**
   Since \( y \) is odd, we know that it can be written as \( y = 2k + 1 \), where \( k \) is some integer.

3. **Step 3: Substitute the expressions for \( x \) and \( y \) into \( 3x + 2y \).**
   \[
   3x + 2y = 3(2j) + 2(2k + 1)
   \]
   Simplify:
   \[
   3x + 2y = 6j + 4k + 2
   \]

4. **Step 4: Factor out a 2.**
   \[
   3x + 2y = 2(3j + 2k + 1)
   \]
   Since \( 3j + 2k + 1 \) is an integer, we have shown that \( 3x + 2y \) is even.

Thus, \( 3x + 2y \) is an even integer.

---

Let me know if you would like a more detailed explanation on any part of this problem.

### Here are 5 related questions:
1. Can you prove that the sum of two even numbers is always even?
2. What happens if both \( x \) and \( y \) are odd? Will \( 3x + 2y \) still be even?
3. How can you prove that the product of an even number and any integer is always even?
4. If \( x \) is odd and \( y \) is even, will \( 3x + 2y \) be odd or even?
5. Can you prove that the sum of an even number and an odd number is always odd?

### Tip:
When working with even and odd numbers, try to always express even numbers as multiples of 2 and odd numbers as multiples of 2 plus 1 for easier proofs.

Let x be an even integer and y be an odd integer. Prove that 3x + 2y is an even integer.

In this problem, we prove that 3x + 2y is an even number when x is an even integer and y is an odd integer. This involves using the properties of even and odd numbers and basic algebraic manipulation. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation