Technique explanation
This lesson covers Merger Equations, utilizing the principle that when numerators sum to a common total, terms can be merged to drop the degree of the equation by one.
A merger occurs when the sum of numerators on the left equals the numerator on the right (A + B = C). By splitting the right-hand term and regrouping, we can cancel a common denominator factor. This transforms a cubic cross-multiplication into a linear equation.
This method applies to rational equations of the form A/(x+a) + B/(x+b) = C/(x+c). If the condition A + B = C is not met, the highest-degree terms will not cancel, and the merger will not work.
Standard fallback: Standard cross-multiplication of all rational terms, which typically results in a cubic equation, followed by the use of the rational root theorem or synthetic division to find solutions.
Worked examples
Example 1: Solve 3/(x+1) + 5/(x+2) = 8/(x+3)
- Step 1: Check 3 + 5 = 8. Correct.
- Step 2: Calculate merged coefficients: A' = 3(3 - 1) = 6, B' = 5(3 - 2) = 5.
- Step 3: Solve 6/(x+1) + 5/(x+2) = 0 → 6(x + 2) + 5(x + 1) = 0 → 11x + 17 = 0.
- Step 4: x = -17/11.
- Check: Substitute x = -17/11 into the LHS: 3/(-17/11 + 1) + 5/(-17/11 + 2) = 3/(-6/11) + 5/(5/11) = -33/6 + 11 = 5.5. RHS: 8/(-17/11 + 3) = 8/(16/11) = 88/16 = 5.5.
Example 2: Solve 1/x + 1/(x+1) + 3/(x+2) = 5/(x+3)
- Step 1: Check 1 + 1 + 3 = 5. Correct.
- Step 2: Merge against c = 3: A' = 1(3 - 0) = 3, B' = 1(3 - 1) = 2, C' = 3(3 - 2) = 3.
- Step 3: Solve 3/x + 2/(x+1) + 3/(x+2) = 0 → 3(x+1)(x+2) + 2x(x+2) + 3x(x+1) = 0.
- Step 4: 8x² + 16x + 6 = 0 → 4x² + 8x + 3 = 0 → (2x+1)(2x+3)=0 → x = -1/2, -3/2.
- Check: Substitute x = -1/2 into the LHS: 1/(-0.5) + 1/(0.5) + 3/(1.5) = -2 + 2 + 2 = 2. RHS: 5/(-0.5+3) = 5/2.5 = 2. Both sides are equal.
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