L3 · Preservation of equality

Learning goal

“I can keep an equation balanced - whatever I do to one side, I do to the other - and use that idea to solve simple equations.”

Preservation of equality — the rule that an equation stays true if the same operation is applied to both sides.

Balance model — picturing an equation as a balance scale: the two sides have equal weight, and the scale must stay level.

Isolate — get the variable alone on one side of the equation, so its value can be read off.

Inverse operation — an operation that undoes another: subtraction undoes addition, division undoes multiplication.

Worked example

Picture x + 3 = 8 as a balance scale. Left pan: a sealed bag (x marbles, unknown count) plus 3 loose marbles. Right pan: 8 marbles. The scale balances - that is what the equals sign means. Step 1: Goal: find how many marbles are in the bag - isolate x. Step 2: Remove 3 marbles from the LEFT pan. The scale tips... unless the same 3 are removed from the RIGHT pan. Do both: left pan now holds just the bag; right pan holds 5. Step 3: The scale still balances, and it says x = 5. Step 4: Check by substitution (L2's skill): 5 + 3 = 8. The claim is true. Say it plainly: you may add, subtract, multiply or divide - but whatever you do to one side, you must do to the other. That is preservation of equality, and it is the only legal move in equation solving.

Check yourself — explain it

Why must the 3 be removed from BOTH pans?

Paired problem — example still visible

A balance shows x + 4 on the left pan and 11 on the right. Describe the preserving move and solve. -> Remove 4 from both pans: x = 7 (check: 7 + 4 = 11)

Faded completion — last step is yours

"m - 5 = 9. Add ___ to both sides; then m = 14" -> 5

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10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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