Lessons · Teaching · solving an equation by undoing it
Solving is undoing, in the reverse of the order it was built
An equation is a balance. Whatever is done to one side must be done to the other, and the way to isolate the unknown is to undo each operation in the reverse order it was applied. Addition is undone by subtraction, multiplication by division.
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In beta. This lesson was written for Hone and has not yet been checked by a state certification officer. Practice material, not professional advice. What that means.
What it is for
A pupil who learns 'move it over and change the sign' has a rule that works and explains nothing, and it fails the moment the equation is unusual. A pupil who sees a balance can check their own answer by substituting it back, which is the habit that makes the rule unnecessary.
How to think about it
Read what has been done to the unknown, in order. Undo the last thing first. Do the same operation to both sides, written out, every line. Substitute the answer back into the original to check, always.
Worked example
3n + 4 = 19: the unknown was multiplied by 3, then 4 was addedName the operations in order before undoing anything.
Undo the addition first: subtract 4 from both sides, giving 3n = 15Last thing done, first thing undone.
Undo the multiplication: divide both sides by 3, giving n = 5Both sides, both times. That is what keeps it balanced.
Check: 3 x 5 + 4 = 19, which matches the originalThe check costs five seconds and catches every sign error.
Your turn
Name what must be done to both sides to keep an equation true.
Whatever you do to one side, do to the
Solve one, graded on the server
The trap
Undoing in the order the operations were written rather than the reverse. Dividing 3n + 4 = 19 by 3 first gives n + 4/3 = 19/3, which is true and useless.