How to Solve Rational Equations

A rational expression is a fraction that has one or more variables in the numerator or denominator. A rational equation is an equation that has at least one rational expression. Like regular algebraic equations, rational equations are solved by performing the same operations on both sides of the equation until the variable is split to one side of the equal sign. The two techniques of cross multiplication and finding the least common denominator are extremely useful for separating variables and solving rational equations.

Table of Contents

Cross multiply

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If necessary, rearrange the equation so that each side of the equal sign has a fraction. Cross multiplication is an easy, fast way to solve rational equations. Unfortunately, this method only works for rational equations that contain only one rational expression or fraction on each side of the equal sign. If the equation is not in canonical cross-multiplication, you may have to use algebraic operations to move its term to the correct canonical position.

For example, the equation (x + 3)/4 – x/(-2) = 0 can be easily reduced to the cross-multiplier form by adding x/(-2) to both sides of the equation, giving (x) + 3)/4 = x/(-2).
- Note that decimals and integers can be converted to fractions by giving them a denominator of 1. For example, (x + 3)/4 – 2.5 = 5 can be rewritten as (x + 3)/4 = 7.5/1 to be able to cross multiply them.
Some rational equations cannot be easily reduced to a form with a fraction or rational expression on each side of the equal sign. In these cases, use the least common denominator method.

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Cross multiply. Cross multiplication simply means multiplying the numerator of one fraction by the denominator of the other. Multiply the numerator of the fraction to the left of the equal sign by the denominator of the fraction to the right of the equal sign. Do the same with the numerator of the fraction on the right and the denominator of the fraction on the left.

Cross multiplication follows basic algebraic principles. Other rational and fractional expressions can be reduced to non-fractional form by multiplying them by their denominator. Cross multiplication is basically a handy shortcut for multiplying both sides of an equation by both denominators of the fraction. Do not believe it? Give it a try – you’ll get the same result after the reduction.

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Set the two products to be equal. After cross-multiplying, you will have two products. Set these two terms equal and reduce so that each side of the equation is in its simplest form.

For example, if your original rational expression was (x+3)/4 = x/(-2), after cross-multiplying, you would get a new equation of -2(x+3) = 4x. If desired, it can also be written as -2x – 6 = 4x.

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Solve the equation to find the variable. Use algebraic operations to solve equations to find variables. Remember, if x occurs on both sides of the equal sign, you will have to add or subtract the term x on both sides so that x is left on only one side.

In this example, we can divide both sides of the equation by -2, the result is x+3 = -2x. Subtracting x on both sides we get 3 = -3x. Finally, dividing both sides by -3 results in -1 = x, or x = -1. We have solved the rational equation to find x.

Find the least common denominator (MSCNN)

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Know when to find the least common denominator. The least common denominator (MSCNN) can be used to simplify rational equations, helping to solve the problem of finding variables. Finding MSCNN is a good idea when your rational equation can’t be easily written in a form where there is one (and only one) rational fraction or expression on each side of the equal sign. To solve rational equations with three or more terms, the MSCNN is a useful tool. However, for a two-term rational equation, cross multiplication is much faster.

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Consider the denominator of each fraction. Determine the smallest number that is divisible by each denominator. This is the MSCNN for your equation.

Sometimes the least common denominator is easy to spot. For example, if your expression is x/3 + 1/2 = (3x+1)/6, it’s not hard to see that the smallest number divisible by 3, 2, and 6 is actually 6.
However, normally the MSCNN of a rational equation is not so easy to find. In these cases, try looking at multiples of the larger denominator until you find a number that has all the smaller denominators as factors. Often MSCNN is a multiple of two of the denominators. For example, in the equation x/8 + 2/6 = (x – 3)/9, MSCNN is 8*9 = 72.
If one or more of the fraction’s denominators contain variables, the process is more complicated, but not impossible. In these cases, MSCNN will be an expression (containing a variable) that is divisible by all denominators. For example, in the equation 5/(x-1) = 1/x + 2/(3x), MSCNN is 3x(x-1), since it is divisible by every denominator – divide it by (x-1) we get 3x, divide it by 3x get (x-1), and divide it by x get 3(x-1).

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Multiplying each fraction in a rational equation by 1. Multiplying each term by 1 sounds pointless. However, this is a trick. 1 can be defined as the quotient of any number divided by itself – like 2/2 and 3/3, for example. This method takes advantage of the above definition. Multiply each fraction in your equation by 1, writing 1 as the number or term that multiplies it by each denominator to get MSCNN on itself.

In the basic example, we would multiply x/3 by 2/2 to get 2x/6 and multiply 1/2 by 3/3 to get 3/6. 3x +1/6 already has 6 as MSCNN, so we can multiply it by 1/1 or keep it the same.
In the example with a variable in the denominator of the fraction, the process is a bit more complicated. Since MSCNN is 3x(x-1), we multiply each rational expression by the term that when multiplied by the denominator yields 3x(x-1) on itself. We’ll multiply 5/(x-1) by (3x)/(3x) to get 5(3x)/(3x)(x-1), multiply 1/x by 3(x-1)/3(x-) 1) get 3(x-1)/3x(x-1), and multiplying 2/(3x) by (x-1)/(x-1) gets 2(x-1)/3x(x-1) .

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Simplify and solve the equation to find x. Now that each term in your equation has the same denominator, you can remove the denominators from the equation and solve the problem in the numerator. Use algebraic operations to find x (or any other variable).

In the basic equation example, after multiplying each term by the alternative form of 1, we get 2x/6 + 3/6 = (3x + 1)/6. Two fractions can add up if they have the same denominator, so we can reduce this equation to (2x + 3)/6 = (3x + 1)/6 without changing its value. . Multiplying both sides by 6 to remove the denominator, we get 2x + 3 = 3x + 1. Subtract 1 on both sides to get 2x + 2 = 3x, and subtract 2x on both sides to get 2 = x , or just x = 2.
In the example of an equation with a variable in the denominator, the new equation after multiplying each term by “1” is 5(3x)/(3x)(x-1) = 3(x-1)/3x(x) -1) + 2(x-1)/3x(x-1). Multiplying each term by MSCNN allows us to remove the denominator, we get 5(3x) = 3(x-1) + 2(x-1). Analyze to 15x = 3x – 3 + 2x -2, then reduce to 15x = x – 5. Subtract x on both sides, we get 14x = -5, finally the result is x = -5/ 14.

Advice

When you’ve solved the problem of finding the variable, check the result by substituting the variable’s value into the original equation. If the value of the variable is true, your original equation will be reduced to the simplest form of 1 = 1.
Note that you can write any polynomial in rational expression form; just put it on the denominator as “1.” So x+3 and (x+3)/1 both have the same value, but the latter is considered a rational expression because it is written as a fraction.

Steps

Cross multiply

Find the least common denominator (MSCNN)

Advice

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