Algebra intro: letters for numbers

1Summary 重點摘要

a + 3 means some number plus 3.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: A letter stands for a numberA letter can hold an unknown number: in a + 3, the letter a could be 5, 10 or any number at all. Writing number relations as expressions with letters makes them short and universal: whatever a is, a + 3 is always 3 more than a, and one line covers every case. This is the first step of algebra. Why use letters? Because some numbers are not yet known, or a rule true for every number needs saying, and a letter is exactly the tool for both. A common misunderstanding is treating a as one fixed value; a letter holds a pending or arbitrary number, and only when the question gives it a value does it become definite.
2. Step 2: Substitute to evaluateOnce a = 5 is known, put 5 in place of a: a + 3 = 5 + 3 = 8. Substituting means swapping every letter for its number, then working out the result in the original order of operations. Every letter in the expression must be replaced, with none left behind. Why follow the order after substituting? Because the expression keeps its own structure, and the rule of multiplying before adding still applies; in 3a + 2, first compute 3 × a, then add 2. A common error is starting to calculate with some letters still unreplaced, or writing the substitution in a tangle; swap out every letter first, then compute step by step.
3. Step 3: Multiplication shorthandIn algebra the multiplication sign is usually left out: a × 5 is written 5a, and a × b is written ab. The number goes in front of the letter, so 3 × a is 3a, never a3. Why may the sign be dropped? Because a number and a letter side by side already mean multiplication, with no confusion with adding or subtracting. But plus and minus signs can never be dropped; only the multiplication sign between a number and a letter may be. Also 1 × a is simply written a, without the 1. A common error is treating a3 as correct, or writing 3 + a as 3a; remember only multiplication may hide, and the number always leads.
4. Step 4: Combine like terms2a + 3a = 5a, because 2 a's plus 3 a's makes 5 a's, just as 2 sweets plus 3 sweets makes 5 sweets. This is called combining like terms. Why may they be combined? Because the two terms share the same letter and so the same unit, letting the coefficients in front add directly. But 2a + 3b cannot become 5ab, because a and b are different numbers, just as 2 sweets plus 3 biscuits cannot be called 5 sweets. The common error is exactly this forced merging of different letters; before combining, check that the letters match completely — combine only when they do, and leave the terms as they are when they do not.
5. Step 5: A simple equationx + 2 = 7 is a simple equation asking: what number plus 2 makes 7? Subtracting gives 7 - 2 = 5, so x = 5. After finding it, a check is required: substitute 5 back, 5 + 2 = 7, both sides equal, so x = 5 is right. Why check? Because the solving process may contain slips, and the check is a reliable confirmation that costs a few seconds, especially worthwhile in tests. A common error is skipping the check, or substituting the wrong number; remember to substitute into the original equation, not into some intermediate line, and only a matching check completes the work.
6. Step 6: A life exampleA life example: buy 3 bottles of water at $w each, and the total is 3w dollars. If the total paid was $15, then 3w = 15, so w = 5, meaning each bottle costs $5. The letter first records the unknown price, the equation then states the relation, and the solution follows — this is exactly what algebra is for: turning word problems into expressions. A common error is swapping the roles of 3 and w, writing w + 3 or w ÷ 3 instead; before writing the expression, be clear about which quantity is the per-item price and which is the total, since price times count gives the total.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• A letter can stand for an unknown number
• Substitute: if a = 5, then a + 3 = 8
• Drop the × sign: 3 × a is written 3a
• Only like terms combine: 2a + 3a = 5a
• Always check by substituting back