Sum, Difference and Special Cases തുകയും വ്യത്യാസവും
When you know the sum (തുക) and the difference (വ്യത്യാസം) of two numbers, one addition and one subtraction finish the job. This page also gathers the textbook's side boxes: what happens when the facts are really the same fact, when the facts cannot all be true, how ancient Chinese mathematicians solved pairs with numbers only, and how a computer does it.
- Two facts about two unknowns give a pair of equations (സമവാക്യജോടി).
- Make one letter's coefficients equal, then subtract – that letter disappears (elimination).
- Substitution: write one letter in terms of the other and put it into the second equation.
1. Sum 28, difference 12 (page 17)
The puzzle: The sum of two numbers is 28 and their difference is 12. What are the numbers? Larger x, smaller y: x + y = 28, x − y = 12.
രണ്ടു സംഖ്യകളുടെ തുകയും വ്യത്യാസവും കൂട്ടിയാൽ വലിയ സംഖ്യയുടെ രണ്ടു മടങ്ങു കിട്ടും. തുകയിൽ നിന്ന് വ്യത്യാസം കുറച്ചാൽ ചെറിയ സംഖ്യയുടെ രണ്ടു മടങ്ങു കിട്ടും.
(x + y) + (x − y) = 2x (x + y) − (x − y) = 2ySum + difference = twice the larger; sum − difference = twice the smaller. So larger = (sum + difference) ÷ 2 and smaller = (sum − difference) ÷ 2.
Push the difference above the sum and watch: the smaller number turns negative. For counts, lengths or prices that is impossible – more on this in section 3.
2. One equation is not enough വിവരങ്ങളും പരിഹാരങ്ങളും
Side box (page 11): a box has 10 beads, black and white. With only this fact, the answer could be 1 black and 9 white, 2 and 8, 3 and 7, … The single equation x + y = 10 has many solutions. Add a second fact, "2 more black than white" (x − y = 2), and only one pair survives: x = 6, y = 4.
Every column adds up to 10 – they all satisfy the first equation. The second equation picks out exactly one column.
3. When the data is impossible കണക്കും കാര്യവും
Side box (page 17): make a rectangle with perimeter 10 m whose length is 5.5 m more than its breadth. Breadth x, length x + 5.5: x + (x + 5.5) = 5 ⇒ 2x = −0.5. A negative breadth! The two conditions cannot both be true for a real rectangle. In other words, x + y = 5 and y − x = 5.5: two positive numbers can never have a difference bigger than their sum.
Algebra will happily give x = −0.25. It is you who must notice that a length, a number of people or a price cannot be negative (or a fraction of a person). Then the honest answer is "no such rectangle exists".
4. Two facts that are really one വ്യത്യസ്തമല്ലാത്ത വിവരങ്ങൾ
Side box (page 13): Ramu paid ₹7 for one pencil and one pen. Anju paid ₹28 for four pencils and four pens. Find each price.
Anju bought exactly 4 times what Ramu bought, and paid exactly 4 times as much. Her bill tells us nothing new: 4x + 4y = 28 is just 4(x + y) = 28, the same as x + y = 7. One real fact, so many possible prices (₹1 and ₹6, ₹2 and ₹5, …).
5. Numbers only – the Chinese table method സംഖ്യകൾ മാത്രം
Side box (page 15): we can drop the letters completely and work with rows of numbers. Ancient Chinese mathematicians solved such problems this way long before algebra. The vessels problem: small vessel 5 times and big vessel 2 times give 20 litres; 2 times and 3 times give 19 litres.
The columns are "small vessel", "big vessel", "water". Every row operation is exactly an operation on an equation – the letters were only labels for the columns!
6. Let the computer do it കമ്പ്യൂട്ടർ ബീജഗണിതം
Programs called Computer Algebra Systems (CAS) – for example the free programs SageMath and Maxima, and GeoGebra – can do algebra, not just arithmetic. GeoGebra and Maxima also run on Android phones. To solve the vessels pair in GeoGebra, open the CAS view (View → CAS) and type:
Solve({5x + 2y = 20, 2x + 3y = 19}, {x, y})
The answer appears as {{x = 2, y = 5}}. The computer is fast, but you still have to turn the problem into the right equations – that part is yours!
7. Three kinds of pairs graph view – beyond the textbook
✅ One solution
Two different facts. Most textbook problems are like this.
♾️ Many solutions
The second fact is the first in disguise.
🚫 No solution
The facts contradict each other.
All the pairs (x, y) that fit one equation lie on a straight line. Two different lines cross at exactly one point – the solution. If the two equations are the same, the lines lie on top of each other (every point fits). If they contradict, the lines are parallel and never meet. You will study graphs properly in later classes.
8. Check yourself
- Sum and difference: larger = (sum + difference) ÷ 2, smaller = (sum − difference) ÷ 2.
- One equation in two letters has many solutions; a pair usually pins down exactly one.
- If one equation is a multiple of the other, the facts are the same: the values cannot be found separately.
- If the facts contradict (or force a negative length), no answer exists – say so.
- The table method and computer algebra do the same elimination, without letters or by machine.