Solving Linear Equations with Fractions

Linear Equations with Fractions

Linear equations are mathematical expressions of relationships between variables where the highest power of each variable is 1. When these equations involve fractions, they can be solved by manipulating the equation to isolate the fraction and reduce it to its simplest form. Here's how you can do this:

Step 1: Get Rid of Fractions

To solve a linear equation involving fractions, your first step is to eliminate them. This usually involves multiplying both sides of the equation by the least common multiple (LCM) of the denominators of all fractions involved. For example, if you have two fractions (x+1)/(x-1) and (-x+2)/(x+3), their LCM would be (x^2 - x + 2)(x^2 + x + 3). By multiplying both sides of the equation by the LCM, the fractions cancel out.

Step 2: Simplify the Fraction

Once you've eliminated the fractions, you may need to further simplify the resulting expression. To do this, you can cross-multiply or distribute and combine like terms as needed. For instance, if you have ((x+1)*(x-1))/(x^2 - x + 2) = (-x+2)*(x+3)/(x^2 + x + 3), cross-multiplying gives you (x^2 - x + 2)((-x+2)*(x+3)) = ((x+1)*(x-1))*((x^2 + x + 3)). Then, distributing and combining like terms results in -x^3 + 2x^2 - 6x + 6 = x^3 - x^2 - x + 2x - 2. Finally, simplifying the left side gives you -x^3 + x^2 + 6x - 4 = 0.

Step 3: Solve the Equation

The last step is to solve the resulting equation. If it's already written in standard form (ax^n + bx^(n-1) + ... = c), you can set it equal to zero and apply any appropriate algebraic methods such as factoring or completing the square depending on the number of variables and the complexity of the coefficients. In our example above, we now have -x^3 + x^2 + 6x - 4 = 0. We can factor this as -(x+2)(x^2 - 3) = 0, leading to the solutions x=-2 and x=sqrt(3).

In conclusion, solving linear equations with fractions involves getting rid of fractions using the LCM, simplifying the resulting expression through cross-multiplication or distribution, and solving the final equation using algebraic techniques.