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Rational equations, also known as fractional equations, are a type of algebraic equation that involve fractions. These equations contain variables in the numerator and/or denominator of one or more fractions. Solving rational equations can be a challenging task, as it requires manipulating fractions and solving for the variables. In this guide, we will explore the step-by-step process of solving rational equations, outlining various strategies and techniques that can be utilized to find the solutions. Whether you are a student studying algebra or an individual seeking to refresh your knowledge, this guide will provide you with the necessary tools to effectively solve rational equations.
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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.
Steps
Cross multiply
- 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.
- 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.
- 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.
- 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)
- 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).
- 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) .
- 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.
wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, volunteer authors have edited and improved the article over time.
This article has been viewed 18,701 times.
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.
In conclusion, solving rational equations involves several key steps to ensure an accurate solution. It is important to simplify the equation by factoring, canceling common factors, and determining any extraneous solutions. Cross-multiplication can be a helpful strategy when solving for variables that appear as denominators. It is crucial to check the final solutions in the original equation to verify their validity. Practice and familiarity with various algebraic techniques, as well as a solid understanding of rational expressions, are essential for successfully solving rational equations. By following these steps and utilizing the appropriate strategies, individuals can confidently tackle and solve rational equations.
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