Guess, check, adjust

The idea

In the trial and error method, different values are substituted for the letter-number until one makes LHS = RHS; this method can be inefficient.

Finding n by guessing

Jasmine wants the matchstick arrangement with exactly 99 sticks. Her equation is 2n + 1 = 99. You cannot see the answer yet, so you will guess a value for *n* and check.

You guess *n* = 10 and the LHS is 21. That is far from 99. What will you do?

• Try a bigger *n*
• Try a smaller *n*
• Keep *n* = 10

Hunt for the solution

Change the guess with the buttons. The bar shows the LHS against the RHS mark. Find the guess that makes them equal, in as few guesses as you can.

Trial and error

The book tried 5, 10, 30, 40, 50 and then 49. Each guess told us if the LHS was too small or too big.

It works, but it can take many guesses. And for 5x − 4 = 7 no whole number works at all: *x* = 2 gives 6 and *x* = 3 gives 11, so the solution lies between them.

In the trial and error method, different values are put in the letter-number until one makes LHS = RHS. This method can be inefficient.

• Guess *n* | LHS 2n + 1 | Compared with 99
• 5 | 11 | too small
• 10 | 21 | too small
• 30 | 61 | too small
• 40 | 81 | too small
• 50 | 101 | too big
• 49 | 99 | equal

Notes

In the trial and error method, we put in different values of the letter-number until one makes LHS = RHS. It can be inefficient: we need a better way.

Check yourself

For 3x + 1 = 22: *x* = 5 gives LHS 16 (too small) and *x* = 10 gives LHS 31 (too big). Which value is the best next guess?

Use trial and error on 4n + 3 = 51. Which *n* makes the two sides equal?

Answer: 12

4 × 12 + 3 = 48 + 3 = 51. (10 gives 43, 11 gives 47, 12 gives 51.)

Why can the trial and error method be inefficient?

You found that 5x − 4 = 7 is solved by a number between 2 and 3. Type it as a decimal (for example 3.5).

Answer: 2.2

5 × 2.2 − 4 = 11 − 4 = 7. So x = 2.2, which is 11/5.

• 7 — correct. Yes! The solution lies between 5 and 10. And 3 × 7 + 1 = 22.
• 2. This is smaller than 5, and 5 was already too small.
• 15. This is bigger than 10, and 10 was already too big.
• It never finds the solution.. It does find it when we guess well. The problem is that it may take very many guesses.
• We may need many guesses, and the solution may not even be a whole number. — correct. Yes! For 5x − 4 = 7 the solution is 2.2, which no whole-number guess can reach.
• Every guess must be correct the first time.. Wrong guesses are fine: each one tells us whether to go up or down. But there can be a lot of them.