# Solving Rational Equations in Algebra

A rational function is a fraction where the numerator and denominator are both polynomials. In the context of algebraic equation solving, a rational equation is an equation of the form

a(x)/b(x) = c(x)/d(x),

where a(x), b(x), c(x), and d(x) are all polynomials. To solve such an equation for x, you must first cross multiply the denominators and transform the original equation into the equivalent equation

a(x)*d(x) = c(x)*b(x), or

a(x)*d(x) - c(x)*b(x) = 0.

Since the left hand side is just another polynomial, solving the original problem just amounts to finding the roots of the polynomial a(x)*d(x) - c(x)*b(x) = 0. Here are some examples.

## Example Problem

Rational equations in which the numerators and denominators of both sides are linear functions are the easiest to solve, since the degree of a(x)*d(x) - c(x)*b(x) is at most 2 (i.e. quadratic).

Example: Alloy A is 10% copper and 90% zinc by weight, Alloy B is 25% copper and 75% zinc, and Alloy C is 50% and 50% zinc. Mary has 100 grams of Alloy A and 200 grams of Alloy B. If she adds equal amounts of Alloy C to both metals, how much of Alloy C does she have to add so that the new metals have the same ratio of copper to zinc?

Solution: Before any of Alloy C is added, the ratio of copper to zinc in Alloy A is 10g/90g, and the ratio of copper to zinc in Alloy B is 50g/150g. (The numerator is the amount of copper and the denominator is the amount of zinc.) If she adds x grams of Alloy C to each metal, the new ratios will be

(10+0.5x)/(90+0.5x) {Alloy A}

(50+0.5x)/(150+0.5x) {Alloy B}

Thus, we need to solve

(10+0.5x)/(90+0.5x) = (50+0.5x)/(150+0.5x), or

(10+0.5x)(150+0.5x) = (50+0.5x)(90+0.5x)

1500 + 80x + 0.25x^2 = 4500 + 70x + 0.25x^2

10x = 3000

x = 300.

So, if Mary adds 300 grams of Alloy C to both metals then they will have the same composition.

## Solution to Problem Shown in Image

(3x + 5) / (x + 1) = (2x - 7) / (4 - x)

(3x + 5)*(4 - x) = (2x - 7)*(x + 1) *[cross multiplying to clear denominators]*

-3x^2 + 7x + 20 = 2x^2 - 5x - 7 *[expanding into quadratic]*

0 = 5x^2 - 12x - 27 *[combining like terms]*

x = 3.815339, -1.415339 *[solving with quadratic formula]*

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