L1 · Equations with rational-number coefficients

Learning goal

“I can solve equations whose coefficients are fractions or decimals - like (2/3)x = 12 or 0.4x = 12 - by undoing with inverse operations, and verify every solution by substitution.”

Rational coefficient — a coefficient that is a rational number: a fraction like 2/3, or a decimal like 0.4, multiplying the variable.

Reciprocal — the flipped fraction: the reciprocal of 2/3 is 3/2. A number times its reciprocal is 1 - which is exactly why reciprocals undo fraction coefficients.

Inverse operation — the operation that undoes another (review from Grade 7): multiplying by 2/3 is undone by multiplying by 3/2.

Verify by substitution — replace the variable with your answer in the ORIGINAL equation and check that the two sides are equal (review from Grade 7; mandatory in this unit).

Worked example

Solve (2/3)x = 12. Step 1: Read the forward story: x was multiplied by 2/3. One arrow only. Step 2: Undo it with the reciprocal: multiply BOTH sides by 3/2. On the left, (3/2) x (2/3) = 1, so x stands alone. Step 3: On the right: 12 x (3/2) = 36/2 = 18. So x = 18. Step 4: Verify in the original: (2/3)(18) = 36/3 = 12. Say it plainly: a fraction coefficient is undone by its reciprocal, because a number times its reciprocal is exactly 1 - the coefficient vanishes, the balance survives. An equally legal route: multiply both sides by the denominator first ((2/3)x x 3 = 2x = 36), then divide by the numerator (x = 18). Same destination; L2 turns that route into a system.

Check yourself — explain it

Why does multiplying (2/3)x by 3/2 leave just x?

Paired problem — example still visible

Solve (1/4)x = 9 -> x = 36 (multiply both sides by 4; check: (1/4)(36) = 9)

Faded completion — last step is yours

"(3/5)x = 15: multiply both sides by the reciprocal 5/3, so x = 15 x (5/3) = ___" -> 25

Ready to practise?

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

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