Content

• To step
• Method 1 of 2: Method One: Cross multiplication
• Method 2 of 2: Method Two: Finding the least common multiple (LCM) of the denominators
• Tips

A rational function is a fraction with one or more variables in the numerator or denominator. A rational equation is any equation that contains at least one rational expression. Like common algebraic equations, rational expressions can be solved by applying the same operation to both sides of the equation until the variable is isolated to one side of the equal sign. Two special methods, cross multiplication and finding the least common multiple of the denominators, are particularly useful for isolating variables and solving rational equations.

To step

Method 1 of 2: Method One: Cross multiplication

1. If necessary, rearrange the equation to make sure that there is a fraction on both sides of the equals sign. Cross multiplication is a fast method of solving rational equations. Unfortunately, this method only works for rational equations that have exactly one rational expression or fraction on both sides of the equal sign. If this is not the case for your equation, then you probably need some algebraic operations to get the terms in the right place.
   • For example, the equation (x + 3) / 4 - x / (- 2) = 0 can easily be converted to the correct cross-multiplication form by adding x / (- 2) to either side of the equation, making the result looks like this: (x + 3) / 4 = x / (- 2).
     • Remember that decimals and integers can be converted to fractions by giving them the denominator 1. (x + 3) / 4 - 2.5 = 5, for example, can be rewritten as (x + 3) / 4 = 7.5 / 1, allowing cross multiplication to be applied.
   • Some rational equations cannot be converted to the correct form that easily. In those cases, use the methods where you use the least common multiple of the denominators.
2. Cross multiplication. Cross multiplication simply means multiplying the numerator of one fraction by the denominator of the other and vice versa. Multiply the numerator of the fraction to the left of the equal sign by the fraction to the right. Repeat with the numerator on the right and the denominator of the fraction on the left.
   • Cross multiplication works according to common algebraic principles. Rational expressions and other fractions can be converted to regular numbers by multiplying the denominators. Basically, cross multiplication is a handy shorthand way of multiplying both sides of the equation by both denominators of the fractions. Do not you believe it? Give it a try - you'll see the same results after simplifying.
3. Make the two products equal to each other. After cross multiplication, you are left with two products. Make these two terms equal and simplify them to get the simplest terms on either side of the equation.
   • For example, if (x + 3) / 4 = x / (- 2) was your original rational expression, then after cross multiplication it becomes equal to -2 (x + 3) = 4x. This can optionally be rewritten as -2x - 6 = 4x.
4. Solve for the variable. Use algebraic operations to find the value of the variable in the equation. Keep in mind that if x appears on both sides of the equal sign, then by adding or subtracting an x ​​term, make sure that there are only x terms on one side of the equal sign.
   • In our example, it is possible to divide both sides of the equation by -2, which gives us x + 3 = -2x. Subtracting x from both sides of the equal sign gives us 3 = -3x. And finally, dividing both sides by -3 we get -1 = x, or also x = -1. Now we have found x that solves our rational equation.

Method 2 of 2: Method Two: Finding the least common multiple (LCM) of the denominators

1. Understand when finding the least common multiple of denominators is obvious. The least common multiple (LCM) of the denominators can be used in simplifying rational equations, making it possible to find the values ​​of their variables. Finding a LCM is a good idea if the rational equation cannot be easily rewritten into a form where there is only one fraction or rational expression on each side of the equals sign. For solving rational equations with three terms or more, LCMs are a useful tool. But for solving rational equations with only two terms, cross multiplication is often faster.
2. Examine the denominator of each fraction. Find the smallest number that is completely divisible by any denominator. This is the LCM of your equation.
   • Sometimes the least common multiple - the smallest number that is completely divisible by each of the denominators - is immediately apparent. For example, if your expression looks like x / 3 + 1/2 = (3x + 1) / 6, it is easy to see that the LCM must be divisible by 3, 2, and 6 and thus equal to 6.
   • But more often the LCM of a rational comparison is not immediately clear at all. In those cases, try the multiples of the largest denominator until you find a number that includes the multiples of the other, smaller denominators. Often the LCM is a product of two denominators. For example, take the equation x / 8 + 2/6 = (x - 3) / 9, where the LCM equals 8 * 9 = 72.
   • If one or more of the denominators contains a variable, this process will be somewhat more difficult, but it is by no means impossible. In those cases, the LCM is an expression (with variables) that fully fits all denominators, not just a single number. As an example, the equation 5 / (x-1) = 1 / x + 2 / (3x), where the LCM equals 3x (x-1), because it is completely divisible by any denominator - division by (x- 1) yields 3x, division by 3x yields (x-1), and division by x yields 3 (x-1).
3. Multiply each fraction in the rational equation by 1. Multiplying each term by 1 may seem useless, but there is a trick here. Namely, 1 can be written as a fraction - eg 2/2 and 3/3. Multiply each fraction in your rational equation by 1, writing 1 each time as the number or term multiplied by each denominator to give the LCM as a fraction.
   • In our example, we can 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 a 6 (lcm) as its denominator, so we can multiply it by 1/1 or just leave it.
   • In our example with variables in the denominators, the whole process is a bit more complicated. Since the LCM equals 3x (x-1), we multiply each rational expression by a fraction that yields 3x (x-1) as the denominator. We multiply 5 / (x-1) by (3x) / (3x) and this gives 5 (3x) / (3x) (x-1), we multiply 1 / x by 3 (x-1) / 3 (x -1) and this gives 3 (x-1) / 3x (x-1) and we multiply 2 / (3x) by (x-1) / (x-1) and this finally gives 2 (x-1) / 3x (x-1).
4. Simplify and solve for x. Now that every term in your rational equation has the same denominator, it is possible to eliminate the denominators from the equation and solve the numerators. Simply multiply both sides of the equation by the LCM to get rid of the denominators so that you are left with only the numerators. Now it has become a regular equation that you can solve for the variable by isolating it on one side of the equal sign.
   • In our example, after multiplying, by using 1 as a fraction, we get 2x / 6 + 3/6 = (3x + 1) / 6. Two fractions can be added if they have the same denominator, so we can write this equation as (2x + 3) / 6 = (3x + 1) / 6 without changing its value. Multiply both sides by 6 to cancel out the denominators, leaving us with 2x + 3 = 3x + 1. Here, subtract 1 from both sides to leave 2x + 2 = 3x and subtract 2x from both sides to leave 2 = x, which can then be written as x = 2 as well.
   • In our example with variables in the denominators, the 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 the LCM makes it possible to cancel out the denominators, which now gives us 5 (3x) = 3 (x-1) + 2 (x-1). Elaborated further, this becomes 15x = 3x - 3 + 2x -2, which can be simplified again as 15x = x - 5. Subtracting x from both sides yields 14x = -5, so that the final answer can be simplified to x = - 5/14.

Tips

• Once you have found the value of the variable, check your answer by entering this value in the original equation. If you get the value of the variable right, you should be able to simplify the equation to a simple, correct theorem, such as 1 = 1.
• Every equation can be written as a rational expression; just place it as a numerator above the denominator 1. So the equation x + 3 can be written as (x + 3) / 1, both have the same value.