Direct Proof: Prove that if 5x - 7 is odd, then 9x + 2 is even

TL;DR

This video explains how to use a direct proof to show that if 5x minus 7 is odd, then 9x plus 2 is even.

Transcript

hi in this video we're going to be doing a proof so let x be an element instead of integers we have to prove that if 5x minus 7 is odd then 9x plus 2 is even so before we do the proof let me just briefly recall what it means for an integer to be even or odd so we'll say x is even and that's the same thing as saying x equals 2k for some integer k so... Read More

Key Insights

• 🦕 Understanding the definitions of even and odd numbers is crucial for solving problems like this.
• 👍 Direct proofs involve assuming a statement is true and proving it to be true based on that assumption.
• 🆘 Manipulating equations can help establish relationships between different variables.
• 😑 Pulling out common factors can simplify expressions and show that they are multiples of a certain number.
• 💭 The thought process behind solving math proofs is as important as the final answer.
• 🎚️ This technique can be used at all levels of mathematics.
• ❓ Mastery of proof techniques is beneficial for tackling more complex mathematical problems.

Questions & Answers

Q: How are even and odd numbers defined in mathematics?

In mathematics, an even number is one that can be expressed as x equals 2k, where k is an integer. Conversely, an odd number is expressed as x equals 2k plus 1, with k being an integer.

Q: What is the strategy used in a direct proof?

A direct proof starts by assuming the statement is true and then seeks to prove it as such. In this case, the assumption is that 5x minus 7 is odd, and the proof aims to demonstrate that 9x plus 2 is even.

Q: How does the proof involve both 5x minus 7 and 9x plus 2?

To involve both equations, the proof writes down 9x plus 2 as 5x minus 7 plus 4x plus 9. By manipulating the equation, it simplifies to 2k plus 10 plus 4x, which can be further simplified as a multiple of 2.

Q: Is there more than one approach to solving this proof?

Yes, there are multiple ways to solve this proof. The technique shown in the video is just one possible method, and there may be other valid approaches as well.

Summary & Key Takeaways

• The video discusses the definitions of even and odd integers.

• The proof assumes that 5x minus 7 is odd and aims to prove that 9x plus 2 is even.

• By manipulating the equations, the proof demonstrates that 9x plus 2 can be written as 2 times an integer, showing that it is even.