Algebraic Proofs Examples

Write a proof for the statement "If 2x – 7 = 13, then x = 10." Which properties are used the proof?

Just like in a proof, we start out with what's given. For us, that would be 2x – 7 = 13. The next step would be adding 7 to both sides (gracias, addition property) and then dividing by 2 (merci, division property). Will that give us the right answer? You betcha. And here's the proof.

Find the error in the following "proof" of the statement, "If 3 + z² = 28, then z = 5." Then, find a counterexample to the statement itself.

The way we can find the error in the proof is if we go through it line by line. It might sound tedious, but thankfully, there are only three lines. The first one is the given equation. Nothing wrong with that.

The second is subtracting 3 from both sides. That seems right too, since 3 – 3 = 0 and 28 – 3 = 25. No problems there.

That must mean the error is in the third line. But the square root of 25 is 5, so where's the problem? Well, z = 5 is only one of the answers. Couldn't z = -5 be an answer too? Yes, it could. That's what the proof didn't consider.

Write a proof for the statement "If x³ + y = 10 and 3(3x – 3) = 9, then x = y."

In other words, the first two equations are given and the last one is what we're trying to prove. We can do it.