Content

• Steps
• Method 1 of 2: Cross-multiplication
• Method 2 of 2: Least Common Denominator (LCN)
• Tips

If you are given an expression with fractions with a variable in the numerator or in the denominator, then such an expression is called a rational equation. A rational equation is any equation that includes at least one rational expression. Rational equations are solved in the same way as any equations: the same operations are performed on both sides of the equation until the variable is isolated on one side of the equation. However, there are two methods for solving rational equations.

Steps

Method 1 of 2: Cross-multiplication

1. 1 If necessary, rewrite the equation given to you so that on each side there is one fraction (one rational expression); only then can you use the cross-multiplication method.
   • For example, given the equation (x + 3) / 4 - x / (- 2) = 0. Move the fraction x / (- 2) to the right side of the equation to write the equation in the proper form: (x + 3) / 4 = x / (- 2).
     • Keep in mind that decimal and whole numbers can be represented as fractions by putting in the denominator 1. For example, (x + 3) / 4 - 2.5 = 5 can be rewritten as (x + 3) / 4 = 7 , 5/1; this equation can be solved using cross-multiplication.
   • If you cannot rewrite the equation as it should, see the next section.
2. 2 Cross-multiplication. Multiply the numerator of the left fraction by the denominator of the right. Repeat this with the numerator of the right fraction and the denominator of the left one.
   • Cross multiplication is based on basic algebraic principles. In rational expressions and other fractions, you can get rid of the numerator by multiplying the numerators and denominators of the two fractions, respectively.
3. 3 Equate the resulting expressions and simplify them.
   • For example, a rational equation is given: (x +3) / 4 = x / (- 2). After multiplying crosswise, it is written as: -2 (x +3) = 4x or -2x 2 6 = 4x
4. 4 Solve the resulting equation, that is, find "x". If "x" is on both sides of the equation, isolate it on one side of the equation.
   • In our example, you can divide both sides of the equation by (-2) and get: x + 3 = -2x. Move the terms with the variable "x" to one side of the equation and get: 3 = -3x. Then divide both parts by -3 to get the result: x = -1.

Method 2 of 2: Least Common Denominator (LCN)

1. 1 The lowest common denominator is used to simplify this equation. This method is applicable when it is impossible to write a given equation with one rational expression on each side of the equation (and use the cross-multiplication method). This method is used when a rational equation with three or more fractions is given (in the case of two fractions, it is better to use cross-multiplication).
2. 2 Find the lowest common denominator of the fractions (or least common multiple). NOZ is the smallest number that is evenly divisible by each denominator.
   • Sometimes NOZ is an obvious number. For example, if the equation is given: x / 3 + 1/2 = (3x +1) / 6, then it is obvious that the least common multiple for the numbers 3, 2 and 6 will be 6.
   • If the NOZ is not obvious, write down the multiples of the largest denominator and find one that will be a multiple of the other denominators. Often, the NOZ can be found by simply multiplying the two denominators. For example, if the equation is x / 8 + 2/6 = (x - 3) / 9, then NOZ = 8 * 9 = 72.
   • If one or more denominators contain a variable, then the process becomes somewhat more complicated (but not impossible). In this case, the NOZ is an expression (containing a variable) that is divided by each denominator. For example, in the equation 5 / (x-1) = 1 / x + 2 / (3x) NOZ = 3x (x-1), because this expression is divisible by each denominator: 3x (x-1) / (x-1 ) = 3x; 3x (x-1) / 3x = (x-1); 3x (x-1) / x = 3 (x-1).
3. 3 Multiply both the numerator and denominator of each fraction by the number equal to the result of dividing the NOZ by the corresponding denominator of each fraction. Since you are multiplying both the numerator and the denominator by the same number, you are actually multiplying the fraction by 1 (for example, 2/2 = 1 or 3/3 = 1).
   • So in our example, multiply x / 3 by 2/2 to get 2x / 6, and multiply 1/2 by 3/3 to get 3/6 (you don't need to multiply 3x +1/6 since it the denominator is 6).
   • Proceed in the same way when the variable is in the denominator.In our second example, NOZ = 3x (x-1), so multiply 5 / (x-1) by (3x) / (3x) and get 5 (3x) / (3x) (x-1); 1 / x multiply by 3 (x-1) / 3 (x-1) and get 3 (x-1) / 3x (x-1); 2 / (3x) multiply by (x-1) / (x-1) to get 2 (x-1) / 3x (x-1).
4. 4 Find "x". Now that you've brought the fractions to a common denominator, you can get rid of the denominator. To do this, multiply each side of the equation by a common denominator. Then solve the resulting equation, that is, find "x". To do this, isolate the variable on one side of the equation.
   • In our example: 2x / 6 + 3/6 = (3x +1) / 6. You can add two fractions with the same denominator, so write the equation as: (2x + 3) / 6 = (3x + 1) / 6. Multiply both sides of the equation by 6 and eliminate the denominators: 2x + 3 = 3x +1. Solve and get x = 2.
   • In our second example (with a variable in the denominator), the equation looks like (after reduction to a common denominator): 5 (3x) / (3x) (x-1) = 3 (x-1) / 3x (x-1) + 2 (x-1) / 3x (x-1). By multiplying both sides of the equation by the NOZ, you get rid of the denominator and get: 5 (3x) = 3 (x-1) + 2 (x-1), or 15x = 3x - 3 + 2x -2, or 15x = x - 5 Solve and get: x = -5/14.

Tips

• Once you've found the x, check your answer by plugging the x value into the original equation. If the answer is correct, you can simplify the original equation to a simple expression such as 1 = 1.
• Note that you can write any polynomial as a rational expression by simply dividing it by 1. So x +3 and (x +3) / 1 have the same meaning, but the last expression is considered a rational expression because it is written as a fraction.