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NCERT Solutions for Class 7th Maths Chapter 7 Generating Equations — Math Talk

Book page 179 – 180 Updated on2026-09-19

Q1.
Write equations whose solution is y = 5. Share the equations you made with each other and discuss the methods used.
Answer

Start from y = 5 and do the same thing to both sides — every step gives a new equation with the same solution.

Operation applied to y = 5Equation you getCheck with y = 5
Add 1y + 1 = 65 + 1 = 6 ✓
Multiply by 33y = 1515 = 15 ✓
Multiply by 3, then add 63y + 6 = 2115 + 6 = 21 ✓
Subtract from 1212 − y = 712 − 5 = 7 ✓
Divide by 5y ÷ 5 = 15 ÷ 5 = 1 ✓
Multiply by 4, then subtract y4y = 3y + 520 = 15 + 5 ✓
Why it happens: this is solving run backwards. When you solve, you strip operations away until only y is left. When you generate, you build operations on until the equation looks interesting. Because every step is done to both sides, the value y = 5 keeps satisfying every equation in the chain.
Try This: make one whose unknown appears on both sides, such as 4y = 3y + 5, and give it to a friend to solve.
Q2.
Can you form a chain going from the bottom equation to the top? [The chains shown are: y + 1 = 6 → (multiplying by −1) −y − 1 = −6 → (adding y) −1 = −6 + y → (adding 6) 5 = y; and 3y = 15 → (adding 6) 3y + 6 = 21 → (dividing by 3) y + 2 = 7 → (subtracting 2) y = 5.]
Answer

Yes — climb back up by applying the inverse operation at each step, taken in the reverse order.

Second chain, bottom to top
y = 5
Add 2 to both sides:   y + 2 = 7
Multiply both sides by 3:   3y + 6 = 21
Subtract 6 from both sides:   3y = 15 ✓
First chain, bottom to top
5 = y
Subtract 6 from both sides:   −1 = y − 6, that is, −1 = −6 + y
Subtract y from both sides:   −1 − y = −6, that is, −y − 1 = −6
Multiply both sides by −1:   y + 1 = 6 ✓
Why it happens: going down the chain you added 6; to come back up you subtract 6. Going down you divided by 3; to come back up you multiply by 3. And the order must be reversed, because the last thing you did on the way down is the first thing you must undo on the way up — just like taking off shoes and socks.
Tip: multiplying by −1 is its own inverse. Doing it twice brings you back to where you started.
Q3.
Compare the operations used when going from the top to the bottom and from the bottom to the top.
Answer

They are inverse operations, in the opposite order.

ChainTop → bottomBottom → top
3y = 15 … y = 5add 6, divide by 3, subtract 2add 2, multiply by 3, subtract 6
y + 1 = 6 … 5 = ymultiply by −1, add y, add 6subtract 6, subtract y, multiply by −1
  • "Add 6" is undone by "subtract 6"; "divide by 3" is undone by "multiply by 3".
  • The list is read backwards, because the last step down is the first step up.
  • Multiplying by −1 is special: it undoes itself.
Why it happens: every operation in these chains is reversible, and reversing a sequence of reversible steps means reversing each step and the order. That is exactly what happens when you solve an equation — solving is nothing but climbing back up a chain that somebody else built.
Q4.
Without calculating, can you find the value of the unknown in each equation in the chains above? [Hint: We have seen that the value that satisfies an equation also satisfies the new equation obtained by performing the same operation on both sides of the original equation.]
Answer

Yes — every equation in both chains has the same solution, y = 5.

EquationPut y = 5True?
y + 1 = 65 + 1 = 6Yes
−y − 1 = −6−5 − 1 = −6Yes
−1 = −6 + y−6 + 5 = −1Yes
5 = y5 = 5Yes
3y = 153 × 5 = 15Yes
3y + 6 = 2115 + 6 = 21Yes
y + 2 = 75 + 2 = 7Yes
y = 55 = 5Yes
Why it happens: each equation in a chain was built from the one above it by doing the same operation to both sides. If y = 5 made the two sides equal before, it must make them equal after — the operation treated both sides alike. So you can read the answer off the bottom of the chain and know it fits every line above.
Tip: this is why writing a chain of equations is a safe way to solve. As long as every step is done to both sides, the answer never changes.
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