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Algebra · Algebraic fractions

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Equations with algebraic fractions

Why multiplying through by the denominators turns a fraction equation into a polynomial one, when the result is linear versus quadratic, and the grade-9 pattern where 1/x-terms breed a quadratic.

Algebra · Algebraic fractions

Equations with algebraic fractions

Why multiplying through by the denominators turns a fraction equation into a polynomial one, when the result is linear versus quadratic, and the grade-9 pattern where 1/x-terms breed a quadratic.

Why it works

An equation with xx in a denominator clears the same way numeric fraction equations did: multiply every term by the denominators, watch them cancel, and solve what's left. The only novelty is that the multiplier now contains xx.

One denominator: one multiplication. 3x+4=5\frac{3}{x} + 4 = 5: subtract 4 first (3x=1\frac{3}{x} = 1), or multiply through by xx (3+4x=5x3 + 4x = 5x). Either way x=3x = 3.

A fraction each side: cross-multiply, brackets compulsory.

3x+1=52x−1  ⇒  3(2x−1)=5(x+1)  ⇒  6x−3=5x+5  ⇒  x=8.\frac{3}{x+1} = \frac{5}{2x - 1} \;\Rightarrow\; 3(2x - 1) = 5(x + 1) \;\Rightarrow\; 6x - 3 = 5x + 5 \;\Rightarrow\; x = 8.

Two xx-denominators can breed a QUADRATIC. Multiply 2x+3x+1=1\frac{2}{x} + \frac{3}{x+1} = 1 through by x(x+1)x(x + 1) — every term, including the 1:

2(x+1)+3x=x(x+1)  ⇒  5x+2=x2+x  ⇒  x2−4x−2=0.2(x + 1) + 3x = x(x + 1) \;\Rightarrow\; 5x + 2 = x^2 + x \;\Rightarrow\; x^2 - 4x - 2 = 0.

The right side became x×(x+1)x \times (x+1) — degree two — so the cleared equation is a quadratic. Solve it with the full toolkit: this one doesn't factorise, so the formula gives x=2±6x = 2 \pm \sqrt{6} (4.454.45 or −0.45-0.45 to 2 d.p.). A factorisable cousin: 1x+1x+2=34\frac{1}{x} + \frac{1}{x+2} = \frac{3}{4} clears (×4x(x+2)4x(x+2)) to 3x2−2x−8=0=(3x+4)(x−2)3x^2 - 2x - 8 = 0 = (3x + 4)(x - 2), so x=2x = 2 or x=−43x = -\frac{4}{3}.

Expect two solutions and check both in the ORIGINAL equation — substitution also confirms neither makes a denominator zero (a value that does must be rejected). x=2x = 2: 12+14=34\frac{1}{2} + \frac{1}{4} = \frac{3}{4}. ✓

Read the demand line. "Give your answers correct to 2 decimal places" signals a formula finish; "show that ... = 0" wants the clearing steps in full before any solving.