Solving One Equation: the Balance സമവാക്യം നിർധാരണം
Every method in this chapter ends the same way: two equations are turned into one equation with one letter, which you then solve. This page makes sure that last step is easy, using the oldest picture in algebra – a balance that must stay level.
- An equation (സമവാക്യം) says two expressions are equal, like the two pans of a level balance.
- 3x + 5x = 8x; 2(x + 1) = 2x + 2; a minus before a bracket flips the signs inside.
1. The golden rule: do the same to both sides
A level balance stays level if you add the same weight to both pans, take the same weight from both, or double (or halve) both. So you may add, subtract, multiply or divide both sides by the same number (never divide by 0). Use this to get x alone on one side.
2. More equations from the chapter
In 2x + 3 = 3x there is an x on each side. Take 2x away from both sides: 3 = x. It does not matter which side the letter ends up on – 3 = x and x = 3 say the same thing.
3. Build and solve your own equation
Choose a, b and a secret x; the page works out c and shows the two moves that recover x: undo the +b, then undo the ×a.
4. Equations with fractions: cross multiplication
The fraction puzzles in the chapter give equations like (x + 1)/y = ½. Two fractions are equal when "top × other bottom" match: (x + 1) × 2 = y × 1, so y = 2(x + 1). Same with x/(y + 1) = ⅓: 3x = y + 1.
5. Always check your answer
Put the answer back into the original equation. If both sides come out equal, it is right; if not, look for a slip.
"A taxi charges ₹50 to start and ₹15 per km; the fare was ₹200. How far?" is 50 + 15x = 200: take 50 from both sides (15x = 150), divide by 15 (x = 10 km). Every bill, recipe or travel question with one unknown is solved this way.
6. Check yourself
- Do the same to both sides: add, subtract, multiply, divide (not by 0).
- Undo in reverse order: first remove what was added, then what was multiplied.
- Letters on both sides: move them to one side first (2x + 3 = 3x ⇒ x = 3).
- Brackets first: 2(x + 1) + 1 = 2x + 3. Fractions: cross multiply ((x + 1)/y = ½ ⇒ y = 2(x + 1)).
- Check by putting the answer back into the original equation.