Word problems with algebra

1Summary 重點摘要

Word problems in three moves: define the unknown, translate into an expression, solve and answer. Algebra is the translator.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What is a word problem?A word problem describes a mathematical situation in sentences, such as 'Mary has 3 apples, her mother gives her some more, and now she has 7 — how many did her mother give?' The quantities hide inside the words, so algebra is used to write the relationship down before solving. The power of algebra is using x to stand for the unknown and turning words into a compact equation; the harder the problem, the more this method pays off, because the approach never changes.
2. 2. Step 1: Define variableStep one is defining the variable: set the quantity the question asks for as x and state exactly what x means. Since the question asks how many the mother gave, write 'let the number the mother gave be x'. The definition must be concrete — not merely 'x is apples', which names an object rather than a quantity, but 'x is the number of apples'. Every later step depends on this definition, so a vague definition corrupts the whole solution.
3. 3. Step 2: Set up equationStep two is writing the equation by translating the sentence piece by piece. Mary had 3, the mother gave x, and now there are 7, so 3 + x = 7 is the equation. Words map onto operations: 'gave' means addition, 'used' means subtraction, 'times' means multiplication. Beginners can jot each phrase as a short sentence first. A wrong equation is the worst error — perfect arithmetic cannot repair it — so read the finished equation back against the problem before solving.
4. 4. Step 3: Solve equationStep three is solving the equation by applying the balance rule to isolate x. From 3 + x = 7, subtract 3 from both sides to get x = 7 − 3 = 4, so the mother gave 4 apples. Whatever is done to one side is done to the other, and a term changes sign when it moves across. The classic slip is subtracting on the left only, yielding x = 7; after each line, confirm both sides remain equal — keeping the balance-scale picture in mind makes the rule hard to forget.
5. 5. Step 4: CheckStep four is checking: substitute the solved value back into the original equation. Put x = 4 into 3 + x: 3 + 4 = 7 matches the right side, so the solution is correct. The check costs one line and catches nearly every arithmetic slip — a copied digit or a missed sign change appears instantly. Even better, substitute into the words themselves: with 4 apples given, 3 + 4 really does make 7, so the situation and the answer agree completely.
6. 6. Step 5: Write answerStep five is writing the answer with all three parts: the number, the unit, and a full sentence answering the question. Here the answer is 'the mother gave 4 apples', not a bare x = 4. Units come from the question: 'how many apples' means apples, 'how many dollars' means dollars. Omitting the unit or leaving only the value of x loses marks, and a full sentence has a bonus effect — if it reads badly, such as 'the mother gave −2 apples', the error in understanding or arithmetic shows at once.
7. 7. Common type: multipleMultiple problems are a standard type where two quantities are linked by a multiplier. 'Dad is 35, which is 5 times John's age — how old is John?' Let John be x, write 5x = 35, solve to x = 7, and check 5 × 7 = 35, so John is 7. Note that 'is 5 times John's age' gives 5x, while 'is 5 years older than John' gives x + 5; confusing multipliers with differences is the classic age-problem error, so circle the words 'times' or 'more' in the question before forming the equation.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Word problems use algebra
• Step 1: Define variable (Let x = ?)
• Step 2: Set up equation
• Step 3: Solve equation
• Step 4: Check + write answer