L2 · Converting to an equivalent integer equation

Learning goal

“I can convert an equation with rational coefficients into an equivalent equation with integer coefficients by multiplying EVERY term by the LCM of the denominators (or by a power of 10 for decimals), then solve the integer equation and verify the solution in the original.”

Equivalent equation — an equation with exactly the same solution as the original. Multiplying every term of an equation by the same non-zero number produces an equivalent equation.

Integer equation — an equation whose coefficients (and constants) are all integers - the destination form of this lesson's conversions.

LCM of the denominators — the least common multiple of all denominators in the equation; the smallest multiplier that clears every fraction at once.

Clearing fractions — multiplying through so that every fractional coefficient becomes an integer.

Worked example

Solve (1/2)x + 3 = 5. (This is the curriculum's own example of an equivalent equation.) Step 1: The only denominator is 2, so the clearing multiplier is 2. Step 2: Multiply EVERY term by 2 - the x-term, the +3, and the 5: (1/2)x x 2 = x; 3 x 2 = 6; 5 x 2 = 10. The equivalent integer equation is x + 6 = 10. Step 3: Solve the integer equation: x = 4. Step 4: Verify in the ORIGINAL (the converted equation's answer must satisfy the original - that is what "equivalent" promises): (1/2)(4) + 3 = 2 + 3 = 5. Say it plainly: scaling a balance by the same factor on both pans changes nothing about what balances it. But the stamp must land on every term - a term that escapes the multiplier breaks the equivalence.

Check yourself — explain it

Why must the +3 be multiplied by 2 as well, when it has no fraction in it?

Paired problem — example still visible

Convert and solve (1/3)x + 2 = 6 -> multiply by 3: x + 6 = 18, so x = 12 (check: (1/3)(12) + 2 = 4 + 2 = 6)

Faded completion — last step is yours

"(1/4)x + 5 = 9, multiplied through by 4, becomes x + 20 = ___" -> 36

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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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