Fractional linear equations

Clear all fractional coefficients

For x/3 + (x-2)/4 = 3, multiply every term by 12. This gives 4x + 3(x-2) = 36. Expand and simplify: 7x-6 = 36, so x = 6. Check the original: 6/3 + (6-2)/4 = 2 + 1 = 3.

Multiplying only the first fraction by 12 would change the equality. The denominator 4 divides the whole numerator x-2, so keep that bracket while multiplying.

A variable denominator adds a restriction

For 3/(x-2) = 6, first require x ≠ 2. Multiply by x-2 on this allowed domain: 3 = 6(x-2). Therefore 3 = 6x-12, so x = 15/6 = 2.5. It is allowed, and 3/(2.5-2) = 6.

An excluded value cannot be made valid by multiplying away its denominator. Record restrictions before transforming the equation, then substitute the final candidate in the original.

Check your understanding 1 The numerator is a whole expression

Solve (x + 1)/2-(x-3)/5 = 4.

Your answer or drawing on paper

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Multiply by 10: 5(x + 1)-2(x-3) = 40. Thus 5x + 5-2x + 6 = 40, 3x = 29 and x = 29/3. Substitution gives (16/3)-(4/3) = 4. The minus sign applies to the whole second bracket.

A context also restricts values

A fixed $36 cost is shared equally among n people, and each pays $9. Find n and state a domain suitable for the story.