Solving the Equation: '{2(x+5)} / {(x + 5)(x - 2)} = {3(x - 2)} / {(x - 2)(x + 5)} + 10 / {(x + 5)(x - 2)}'

As a high school mathematics teacher, I often encounter complex algebraic equations that challenge my students. One such equation is '{2(x+5)} / {(x + 5)(x - 2)} = {3(x - 2)} / {(x - 2)(x + 5)} + 10 / {(x + 5)(x - 2)}'. In this article, I will guide you through the step-by-step process of solving this equation.

Step 1: Understand the Equation

The given equation is a complex rational equation, where both the left-hand side and the right-hand side are fractions. To solve this equation, we need to manipulate the fractions and isolate the variable 'x'.

Step 2: Simplify the Fractions

Let's start by simplifying the fractions on both sides of the equation.

On the left-hand side: {2(x+5)} / {(x + 5)(x - 2)} = 2(x+5) / [(x + 5)(x - 2)]

On the right-hand side: {3(x - 2)} / {(x - 2)(x + 5)} + 10 / {(x + 5)(x - 2)} = [3(x - 2)] / [(x - 2)(x + 5)] + 10 / [(x + 5)(x - 2)]

Step 3: Find a Common Denominator

To simplify the equation further, we need to find a common denominator for all the fractions. The common denominator is: (x + 5)(x - 2)

Step 4: Rewrite the Equation with the Common Denominator

Rewriting the equation with the common denominator, we get: [2(x+5)] / [(x + 5)(x - 2)] = [3(x - 2)] / [(x - 2)(x + 5)] + [10(x - 2)] / [(x + 5)(x - 2)]

Step 5: Simplify the Equation

Simplifying the equation, we get: 2(x+5) = 3(x - 2) + 10(x - 2) 2x + 10 = 3x - 6 + 10x - 20 2x + 10 = 13x - 26 -11x = -36 x = 3.27 (rounded to two decimal places)

Conclusion

By following the step-by-step process, we have successfully solved the complex algebraic equation '{2(x+5)} / {(x + 5)(x - 2)} = {3(x - 2)} / {(x - 2)(x + 5)} + 10 / {(x + 5)(x - 2)}' and found the value of 'x' to be approximately 3.27. This type of equation-solving practice is crucial for high school mathematics students to develop their problem-solving skills and prepare for more advanced mathematical concepts.