Two Equations രണ്ടു സമവാക്യങ്ങൾ
In Section 1 we combined two equations by substitution. Here is a second, often quicker way. If both equations contain the same number of one item, subtract them and that item disappears. If they don't, multiply first to make the numbers match. This is called elimination – making one letter vanish.
- Doing the same thing to both sides keeps an equation true – including multiplying both sides by 2 or 5.
- Two letters need two facts: each fact becomes an equation (സമവാക്യം).
- Minus a bracket changes every sign inside: −(12x − 8y) = −12x + 8y.
1. Pens and notebooks
The puzzle (page 11): 2 pens and 3 notebooks cost ₹110. 2 pens and 5 notebooks cost ₹170. What does a pen cost? A notebook?
🧠 In your head
The second bill is ₹60 more because it has 2 more notebooks. So a notebook costs ₹30. Then 2 pens cost 110 − 90 = 20: a pen costs ₹10.
✏️✏️ Two equations
If one letter has the same coefficient (the number in front of it) in both equations, subtract one equation from the other. That letter disappears and only one letter is left.
2. Pencils and pens – make the numbers match
The puzzle (page 12): 3 pencils and 4 pens cost ₹66. 6 pencils and 3 pens cost ₹72. Find the price of each.
Now neither count matches: 3 pencils against 6, 4 pens against 3. But we can buy the first basket twice: 6 pencils and 8 pens cost ₹132.
If the coefficients are different, multiply one equation (or both) by suitable numbers so that one letter gets the same coefficient in both. Then subtract. Multiplying a whole equation by a number keeps it true – it is just the same fact for twice (or five times) the basket.
3. Vessels of water – multiply both
The puzzle (page 14): Filling a small vessel 5 times and a big vessel 2 times and emptying them into a tank gives 20 litres. Small 2 times and big 3 times gives 19 litres. How much does each vessel hold?
To make 5x into 2x you could multiply the first equation by ⅖ – it works, but brings fractions. Multiplying the first by 2 and the second by 5 makes both x-coefficients 10, with whole numbers all the way.
4. Watch the minus sign
The puzzle (page 16): 4 times one number plus 3 times another is 43. 3 times the first minus 2 times the second is 11. Find the numbers.
(12x + 9y) − (12x − 8y) is 17y, not y. Taking away "−8y" is the same as adding 8y.
5. The elimination machine
The right-hand totals are worked out from your secret x and y, so the answer always comes out whole. Multiply equation (1) by a₂ and equation (2) by a₁ (or by smaller numbers if they share a factor) and the x-terms match.
6. One equation is not enough വിവരങ്ങളും പരിഹാരങ്ങളും
Side box (page 11): A box has 10 beads, black and white. With only this fact, can you say how many are black? No – 1 and 9, 2 and 8, 3 and 7 … all work. Add a second fact, "2 more black than white", and only one pair is left.
One equation in two letters has many solutions (പരിഹാരങ്ങൾ). A pair of equations usually has exactly one.
7. A picture of a pair of equations beyond the textbook
Each equation in x and y has many solutions. Plot them all and they lie on a straight line. Two equations give two lines – and the point where they cross is the one pair that fits both. (Graphs of equations come in higher classes; this is just a peek.)
Two shopping bills, two recipes, two tank-filling trips: whenever two mixtures use the same two things in different amounts, elimination finds the price, size or speed of each. Computers solve thousands of such equations at once by exactly this row-by-row method.
8. Check yourself
- Same coefficient? Subtract the equations: 2x + 5y = 170 minus 2x + 3y = 110 gives 2y = 60.
- Different coefficients? Multiply first: ×2 turns 3x + 4y = 66 into 6x + 8y = 132, matching 6x + 3y = 72.
- Multiply both if needed (×2 and ×5), and keep whole numbers.
- Watch signs: (12x + 9y) − (12x − 8y) = 17y.
- After finding one letter, put it back into either equation to find the other – and check in both.
- One equation in two letters has many answers; a pair usually pins down one.