Equations with rational coefficients are equations in which the coefficients are fractions. They can be difficult to solve, but with the right approach, they are manageable. In this lesson, we’ll discuss how to solve equations with rational coefficients and provide some examples so you can practice.
Understanding Rational Coefficients
Rational coefficients are fractions. They can be written as either mixed numbers (a combination of a whole number and a fraction) or as improper fractions (a fraction with a numerator larger than the denominator). In either case, the equation will have to be simplified before it can be solved.
Solving Equations with Rational Coefficients
The first step in solving equations with rational coefficients is to simplify the equation. This can be done by multiplying both sides of the equation by the least common denominator of the fractions in the equation. Once the equation has been simplified, it can be solved like any other equation.
The second step involves isolating the variable. To do this, use inverse operations to move all the terms that are not the variable to one side of the equation and all the terms that are the variable to the other side. Once the variable is isolated, it can be solved.
Examples
Let’s look at a few examples. The first equation is: 2/3x – 4/5 = 1/6. To solve this equation, start by multiplying both sides of the equation by the least common denominator of the fractions, which is 15. The equation now becomes 10x – 12 = 5. Then, use inverse operations to isolate the variable. To do this, add 12 to both sides of the equation. The equation now becomes 10x = 17. Finally, divide both sides by 10 and the answer is x = 17/10.
The second example is: 3/4x + 5/8 = 1/2. The first step is to simplify the equation by multiplying both sides by the least common denominator of the fractions, which is 8. The equation now becomes 6x + 5 = 4. To isolate the variable, use inverse operations. Subtract 5 from both sides and the equation becomes 6x = -1. Finally, divide both sides by 6 and the answer is x = -1/6.
Solving equations with rational coefficients can be challenging, but with the right approach, it is manageable. The first step is to simplify the equation by multiplying both sides by the least common denominator of the fractions. The second step is to isolate the variable by using inverse operations. Once the variable is isolated, it can be solved. With practice, you’ll become more comfortable with solving equations with rational coefficients.
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