Making equations, and mending mistakes

So far we have solved a given equation to find the letter-number. Can we do the REVERSE — write equations from a given value?

Write equations whose solution is y = 5. Two easy ones are y + 1 = 6 and 3y = 15. Share yours with a classmate and discuss the methods used; there are far more than you would guess.

Now build a CHAIN, where each equation comes from the one before by performing the same operation on both sides. Starting from y + 1 = 6: multiply by −1 to get −y − 1 = −6; add y to get −1 = −6 + y; add 6 to get 5 = y. Starting from 3y = 15: add 6 to get 3y + 6 = 21; divide by 3 to get y + 2 = 7; subtract 2 to get y = 5.

Here is the point of the chains. The value that satisfies an equation also satisfies the new equation made by doing the same operation to both sides. So WITHOUT calculating, you already know the unknown in every equation of a chain — it is the same all the way down. Try forming a chain from the bottom equation back to the top, and compare the operations each way.

Example 13: can you give a real-life situation modelled by 100x + 75 = 250? Thinking of the numbers as money, the total is ₹250; the ₹75 is fixed and does not change; the first term changes with x. So 100 could be a number of units and x the cost per unit — or x could be the cost of a plate of snacks and 75 the delivery charge.

Last, mind the Mistake, Mend the Mistake. Here are solutions with their steps; go through each, decide whether the steps are correct, and if not, describe the mistake, correct it and solve properly.

Look at 4x + 6 = 10 'solved' as 4x = 10 + 6. The term 6 was moved but kept its sign, when it must become its additive inverse: 4x = 10 − 6, so 4x = 4 and x = 1. Look at 3x + 1 = −12 'solved' as x + 1 = −12/3. Only ONE term of the LHS was divided by 3 — dividing a side means dividing ALL of it. These are the two mistakes to watch for in your own work, which is why checking the solution matters.