For instance, the equation (x + 3)/4 - x/(-2) = 0 can easily be rearranged into cross-multiplication form by adding x/(-2) to both sides of the equation, leaving you with (x + 3)/4 = x/(-2). Keep in mind that decimals and whole numbers can be made into fractions by giving them a denominator of 1. (x + 3)/4 - 2. 5 = 5, for instance, can be rewritten as (x + 3)/4 = 7. 5/1, making it a valid candidate for cross-multiplication. Some rational equations can’t easily be reduced into a form with one fraction or rational equation on each side of the equals sign. In such cases, use a lowest common denominator approach.
Cross-multiplication works according to basic algebraic principals. Rational expressions and other fractions can be made into non-fractions by multiplying them by their denominators. Cross-multiplication is basically a handy shortcut for multiplying both sides of the equation by both fraction’s denominators. Don’t believe it? Try it - you’ll get the same results after simplifying.
For example, if your original rational expression was (x+3)/4 = x/(-2), after cross multiplying, your new equation is -2(x+3) = 4x. If we wish, this can also be written as -2x - 6 = 4x.
In our example, we can divide both sides of the equation by -2, giving us x+3 = -2x. Subtracting x from both sides gives us 3 = -3x. Finally, dividing both sides by -3 gives us -1 = x, which we can re-write as x = -1. We have found x, solving our rational equation.
Sometimes the lowest common denominator - that is, the lowest number that has each of the existing denominators as a factor - is obvious. For example, if your expression is x/3 + 1/2 = (3x+1)/6, it’s not hard to see that the smallest number with 3, 2 and 6 as a factor, is, in fact, 6. Often, however, a rational equation’s LCD isn’t immediately obvious. In these cases, try examining multiples of the larger denominator until you find one that contains all of the smaller denominators as a factor. Often, the LCD is a multiple of two of the denominators. For example, in the equation x/8 + 2/6 = (x - 3)/9, the LCD is 8*9 = 72. If one or more of your fractions’ denominators contains a variable, this process is more involved, but not impossible. In these cases, the LCD will be an expression (containing variables) that all the denominators divide into, not a single number. For example, in the equation 5/(x-1) = 1/x + 2/(3x), the LCD is 3x(x-1), because each denominator divides into it evenly - dividing it by (x-1) gives 3x, dividing it by 3x gives (x-1), and dividing it by x gives 3(x-1).
In our 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, the LCD, as its denominator, so we can either multiply it by 1/1 or leave it alone. In our example with variables in the denominators of our fractions, the process is slightly trickier. Since our LCD is 3x(x-1), we multiply each rational expression by the term which it multiplies with to give 3x(x-1) over itself. We would multiply 5/(x-1) by (3x)/(3x) giving 5(3x)/(3x)(x-1), multiply 1/x by 3(x-1)/3(x-1) to give 3(x-1)/3x(x-1), and multiply 2/(3x) by (x-1)/(x-1) to give 2(x-1)/3x(x-1).
In our basic example, after multiplying every term by alternate forms of 1, we get 2x/6 + 3/6 = (3x+1)/6. Two fractions can be added together if they have the same denominator, so we can simplify this equation as (2x+3)/6 = (3x+1)/6 without changing its value. Multiply both sides by 6 to cancel the denominators, which leaves us with 2x+3 = 3x+1. Subtract 1 from both sides to get 2x+2 = 3x, and subtract 2x from both sides to get 2 = x, which can be written as x = 2. In our example with variables in the denominators, our 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 our LCD allows us to cancel the denominators, giving us 5(3x) = 3(x-1) + 2(x-1). This works to 15x = 3x - 3 + 2x -2, which simplifies to 15x = x - 5. Subtracting x from both sides gives 14x = -5, which, finally, simplifies to x = -5/14.
