Chapter 1 · RevisionEverything on one page

Summary & Quick Quiz പുനരവലോകനം

All the methods of the chapter in one place, the three kinds of pairs, a word-to-algebra dictionary, the mistakes to avoid, flashcards, a scored quiz and your study checklist. Use it the evening before a test.

1. Three ways to crack a two-unknown puzzle

🧠 Mental

Compare the two facts and ask what the difference pays for. Table + 1 chair = ₹11000, table + 4 chairs = ₹14000 ⇒ 3 chairs cost ₹3000 ⇒ chair ₹1000.

Page 5

✏️ One letter

Name one unknown x and write the other with it: table = 11000 − x. Then one equation: (11000 − x) + 4x = 14000 ⇒ x = 1000.

Page 5 · basics on page 2

✏️✏️ Two letters

Each fact becomes an equation: x + y = 11000, 4x + y = 14000. Then combine them – by substitution or by elimination – until one letter is left.

Page 5 · page 7

2. The methods

Substitution

Use one equation to say what one letter equals, and put that in the other.

y = 2(x + 1), y + 1 = 3x 2(x + 1) + 1 = 3x ⇒ x = 3, y = 8

Fraction puzzle, page 5

Elimination

Same coefficient? Subtract. Different? Multiply first to make them equal.

(1) 3x + 4y = 66 (2) 6x + 3y = 72 2×(1): 6x + 8y = 132 minus (2): 5y = 60 ⇒ y = 12, x = 6

Page 7

Sum & difference

Add the equations, then subtract them.

x + y = 28, x − y = 12 2x = 40 ⇒ x = 20; 2y = 16 ⇒ y = 8 larger = (sum + diff) ÷ 2

Page 9

Table (numbers only)

The ancient Chinese way: rows of coefficients and totals, no letters.

5 2 | 20 ×2 → 10 4 | 40 2 3 | 19 ×5 → 10 15 | 95 subtract → 0 11 | 55 ⇒ 0 1 | 5

Page 9

🧮 Elimination helper – type any pair

x + y =

Use a negative number for a minus sign (3x − 2y = 11 → 3, −2, 11). The helper multiplies the equations to match the x-coefficients, subtracts, and then finds x – exactly the textbook steps.

3. Three kinds of pairs

One solution

x + y = 10, x − y = 2 ⇒ x = 6, y = 4. The usual case: two different facts.

Same information

x + y = 7, 4x + 4y = 28: the second is just 4 times the first. Many answers – the prices cannot be found separately.

Impossible data

x + y = 5, y − x = 5.5 for lengths: x would be negative. Or x + y = 5, 2x + 2y = 12: the facts contradict each other.

The little graphs show each equation as a line (beyond the textbook): crossing lines = one solution, the same line = many, parallel lines = none.

4. From words to algebra

Words	Malayalam	Algebra
the sum of two numbers is 26	തുക	x + y = 26
their difference is 4	വ്യത്യാസം	x − y = 4
the larger is 5 times the smaller	5 മടങ്ങ്	y = 5x
10 more black than white	കൂടുതൽ	x = y + 10
adding 1 to the numerator gives ½	അംശം	(x + 1)/y = ½ ⇒ y = 2(x + 1)
perimeter 40 cm	ചുറ്റളവ്	2(l + b) = 40 ⇒ l + b = 20
tens digit a, units digit b	രണ്ടക്കസംഖ്യ	10a + b; reversed 10b + a
7% interest on x	പലിശ	0.07x
4 years ago / in 2 years	പ്രായം	x − 4 / x + 2
speed in the t-th second	വേഗം	u + at

⚠️ Common mistakes

Sign errors when subtracting: (12x + 9y) − (12x − 8y) = 17y, not y. Subtracting −8y adds 8y.
Multiplying only one side: 2 × (3x + 4y = 66) is 6x + 8y = 132 – the 66 must be doubled too.
Forgetting the second unknown: after y = 12, put it back to find x.
Not checking: substitute your answer in both original equations.
Ignoring sense: a negative length, half a person or a fraction of a coin means the data or the working is wrong.
Mixing up who is bigger: "4 more girls than boys" is g = b + 4, not b = g + 4.

5. Flashcards

Click a card to flip it.

6. Scored quiz

7. Your study checklist

🧭 The whole chapter in four lines

Two unknowns need two facts; write each fact as an equation.
Combine the equations so that one letter disappears – substitution or elimination.
Sum and difference: add and subtract; the table method does the same with numbers only.
Always check, and watch for pairs that say the same thing or say impossible things.

Next: Model test paper ▶