Chapter 15: Finding the Unknown

Sometimes a problem hides a number. "A number, when doubled and added to $7$, gives $19$. What is the number?" You can guess until you hit it , or you can write an equation and solve it.

An equation is just a statement that two expressions are equal. Once we name the unknown with a letter like $x$ or $n$, we can apply a small number of clean rules to isolate it, and out pops the answer. No guessing required.

In this chapter you will learn three big ideas. First, how to build an equation from a real situation , weighing scales, matchstick patterns, word problems. Second, how to solve an equation systematically using the balance principle: whatever you do to one side, do to the other. Third, how to check your answer by substituting back.

These are the first real steps into algebra , a tool that will carry you all the way to physics, computer science and economics. Treat each equation as a puzzle with a guaranteed solution, hidden behind a few moves.

What's inside

1. Equations from real situations , balance scales, sticks, words to symbols.
2. Trial and error: a first method , useful for intuition, slow for precision.
3. The balance method , adding, subtracting, multiplying, dividing both sides.
4. Solving equations systematically , moving terms across the $=$ sign.
5. Equations from word problems , translating English into algebra and back.

Key results

Transposing. A term added on one side moves to the other side subtracted (and vice versa). A factor multiplying one side moves to the other side as a divisor (and vice versa).

Always check. Substitute your value back into the original equation: the LHS should equal the RHS.

How to read this chapter

For every equation you solve, write the verification line below it. You will catch every silly error before it costs you marks , and you will build the habit professional mathematicians and engineers use without thinking.