Trial and error: a first method

The simplest way to solve an equation is to guess a value, plug it in, and see if the two sides match. If not, adjust and try again. This is trial and error (also called the guess and check method).

Idea

Given an equation like $2n + 1 = 99$, pick a value for $n$ and check.

• Try $n = 10$: LHS $= 2(10) + 1 = 21$. Too small.
• Try $n = 50$: LHS $= 101$. A bit too big.
• Try $n = 49$: LHS $= 99$. Match! So $n = 49$.

When to use it. Trial and error is useful when:

1. The equation is simple, with whole-number solutions.
2. You want to build intuition for how the LHS responds to changes in the variable.
3. You only have a small range of possibilities (say, "the age must be between $10$ and $20$").

When NOT to use it. It is terrible when:

1. The answer is a fraction or decimal , you might never hit it by guessing.
2. The numbers are large.
3. The equation involves multiple unknowns or many operations.

Smart guessing. Don't pick random numbers. Use the size of the equation to estimate:

• If $5x = 60$, you know $x$ is around $12$.
• If $x^2 = 169$, you know $x$ is somewhere near $13$.
• If LHS grows fast with $x$, jumping by tens (then ones) gets you close quickly.

Why we still teach it. Trial and error matches the meaning of solving an equation: finding a value that makes both sides agree. Once you grasp this, the balance method (next subtopic) feels like a clever speed-up of the same idea.

Worked examples

Example 1. Solve $3x + 5 = 20$ by trial and error.

Try $x=3$: LHS $= 9 + 5 = 14$. Too small. Try $x=5$: LHS $= 20$. Match! So $x = 5$.

Example 2. Solve $7y - 3 = 32$.

Try $y=5$: $35-3 = 32$. Match. So $y = 5$.

Example 3. Solve $\dfrac{a}{4} = 9$.

Try $a = 36$: $\tfrac{36}{4} = 9$. Match. So $a = 36$.

Example 4. Solve $2n + 1 = 200$ (from the matchstick problem).

Try $n=100$: LHS $= 201$. Off by $1$. Try $n=99.5$: but $n$ must be a whole number, so there is no whole-number solution , you cannot make $200$ matchsticks fit this pattern.

Try it yourself

1. Solve $x + 8 = 15$.
2. Solve $4x = 32$.
3. Solve $5y - 2 = 18$.
4. Solve $3z + 7 = 25$.
5. Solve $\dfrac{m}{6} = 7$.
6. Solve $2x + 1 = 99$ from the triangle-pattern problem.
7. Try to solve $4n + 1 = 50$. Is there a whole-number solution?
8. Solve $9p - 5 = 67$.

Activity

With a partner, play "guess my equation". One of you secretly writes an equation like $3x + 2 = 17$ and tells the other only the answer ($x = 5$). The other player must guess the equation by trying possibilities. Reverse roles. This builds intuition for how LHS depends on $x$.