Acting Rationally with FractionFilled Equations in .NET

Make QR Code in .NET Acting Rationally with FractionFilled Equations

A rational term in an equation is a fraction an equation with one or more terms, all of which are rational, must be a rational equation You probably hope that all your problems (and the people you associate with) are rational, but an equation that contains fractions isn t always easy to handle A general plan for solving a rational equation is to get rid of the fraction or fractions by changing the equation into an equivalent form with the same answer a form that makes it easier to solve

Two of the most common ways to get rid of the fractions are multiplying through by the least common denominator (LCD) and cross-multiplying proportions I just happen to discuss both of these techniques in the sections that follow This mathematical slight of hand using alternate equations to solve more complicated problems isn t without its potential problems At times, the new equation produces an extraneous solution (also referred to as an extraneous root), a false solution that pops up because you messed around with the original format of the equation To guard against including extraneous solutions in your answers, you need to check the solutions you come up with in the original equations Don t worry; I have you covered in the following sections

You can solve rational equations, such as 3x + 2 - 5 = x + 3 , without as 2 2x - 3 4 much hassle if you simply get rid of all the denominators To do so, you work with an old friend, the least common denominator In case you re not on a first-name basis with her, the least common denominator (LCD) is also known as the least common multiple the smallest number that two or more other numbers all divide into evenly (such as 2, 3, and 4 all dividing the LCD 12 evenly) (Head to 18 for a quick trick on finding an LCD) To solve the previous example equation with the LCD, you find a common denominator, write each fraction with that common denominator, and then multiply each side of the equation by that same denominator to get a nice quadratic equation (see 3 for a full discussion of quadratic equations) Quadratic equations have two solutions, so they present more opportunities for extraneous solutions Be on the lookout!

The first step in solving the rational equation is to find the least common denominator (LCD) for all the terms in the equation For example, the common denominator of all three fractions in the equation 3x + 2 - 5 = x + 3 consists of all the factors in the three denominators 2 2x - 3 4 Each of the denominators has to be able to divide into the common denominator evenly In other words, the LCD is a multiple of each of the original denominators To solve this equation, use 4(2x 3) as the common denominator,

because it s a multiple of 2 you multiply by 2(2x 3) to get it; it s a multiple of 2x 3 you multiply by 4 to get it; and it s a multiple of 4 you multiply by (2x 3) to get it All three denominators divide this product evenly

Multiply each of the terms in the original equation by some value so that, after multiplying, each resulting term has the same denominator the LCD you re so fond of: 3x + 2 2 ^ 2x - 3 h - 5 4 = x + 3 2x - 3 2 $ 2 ^ 2x - 3 h 2x - 3 $ 4 4 $ 2x - 3 2 ^ 3x + 2h^ 2x - 3 h ^ x + 3 h^ 2x - 3 h 20 = 4 ^ 2x - 3 h 4 ^ 2x - 3 h 4 ^ 2x - 3 h The some value I speak of is equal to one, because each of the fractions multiplying the terms is the same in the numerator and denominator But you carefully select the fractions that serve as multipliers the numerators and denominators must consist of all the factors necessary to complete the LCD You can just divide the LCD by the current denominator to determine what more you need to create the common denominator in that term