Generating equations with a given solution

The idea

Equations with a given solution can be generated by starting from 'letter-number = value' and performing the same operation on both sides, since a value that satisfies an equation also satisfies each new equation obtained.

Start from y = 5

Suppose we know the answer already: y = 5. Can we write other equations that have 5 as their solution? Try one step. Multiply both sides by 3, and you get 3y = 15.

Is y = 5 also the solution of 3y = 15?

• Yes: 3 × 5 = 15
• No: the solution changes
• Only if we divide again

Build chains of equations

Pick an operation and a number. It is done to both sides. Under every line, y = 5 is put in to see if both sides still agree.

The same solution goes along the chain

Doing the same thing to both sides keeps a balanced scale balanced. So a value that fits the first equation also fits every equation made from it. Each line of the chain keeps the solution y = 5.

That is why you can find the solution of every equation in a chain without calculating: it is the same as the solution of the first one. And the chain can be walked backwards too, by the inverse operations.

To make equations with a given solution, start from letter = value and do the same operation on both sides, again and again. The value that satisfies one equation satisfies every equation made this way.

• Equation | Put in y = 5
• y = 5 | 5 = 5
• 3y = 15 | 3 × 5 = 15
• 3y + 6 = 21 | 15 + 6 = 21
• y + 2 = 7 | 5 + 2 = 7

Notes

Start from letter = value and do the same operation on both sides: every equation you get has that same solution. For example y = 5, 3y = 15, 3y + 6 = 21 all have y = 5.

Check yourself

Start with y = 5. Multiply both sides by 4, then subtract 1 from both sides. Which equation do you get?

Which of these cannot have been made from y = 5 by doing the same thing to both sides?

Start with x = 4. Multiply both sides by 3, then add 2 to both sides. What is the right side of the new equation?

Answer: 14

4 × 3 = 12 and 12 + 2 = 14, so the new equation is 3x + 2 = 14.

A chain starts at m = 8. After some steps it reaches 2m + 10 = 26. Without solving, what is *m* in this equation?

Answer: 8

Every equation in a chain made by the same operation on both sides has the same solution: m = 8. Check: 2 × 8 + 10 = 26.

• 4y − 1 = 19 — correct. Yes! 4 × 5 = 20, and 20 − 1 = 19.
• 4y − 1 = 20. The right side is 5 × 4 = 20, but then 1 is subtracted from it as well: 19.
• 4y + 1 = 19. We subtracted 1, so the left side has − 1, not + 1.
• 2y + 1 = 11. It can: 2 × 5 + 1 = 11. Both sides agree.
• 3y − 5 = 10. It can: 3 × 5 − 5 = 10. Both sides agree.
• y + 4 = 10 — correct. Yes! Put in y = 5: 5 + 4 = 9, not 10. The sides are not equal.