Adding and subtracting rational expressions

Students will learn how to add and subtract rational expressions, with and without like denominators. Students will solve both word problems and mathematical ones. For adding and subtracting with unlike denominators, students will start by learning how to find the LCD of two rational expressions. Sometimes, this will require factoring quadratics. Next, give your students this challenge:

Find the Area of the Yellow Rectangle | 2 Fast & Easy Methods by PreMath
After students have solved the original problem, give them this generalization of the problem: Replace the \(7\) with \(a,\) the \(8\) with \(b,\) the \(28\) with \(A_1,\) and the \(35\) with \(A_2.\) This generalization of the problem can be solved in the same way, but it's a bit more abstract, and thus, more difficult. We start by noting \((a - x)y = A_1,\) and \((b - x)y = A_2.\) Expanding both, we get \(ay - xy = A_1\) and \(by - xy = A_2.\) Subtracting the second from the first, we obtain \(ay - by = A_1 - A_2.\) Solving for \(y\) gives us \(y = (A_1 - A_2)/(a - b).\) Returning to a previous equation, \(ay - xy = A_1,\) we solve for \(xy,\) the area of the middle rectangle, to obtain \(xy = ay - A_1.\) Making a substitution, and simplifying, we get a beautiful formula:

$$\begin{align} xy &= \dfrac{a(A_1 - A_2)}{a - b} - A_1 \\[1em] &= \dfrac{a(A_1 - A_2)}{a - b} - \dfrac{A_1(a - b)}{a - b} \\[1em] &= \dfrac{aA_1 - aA_2 - aA_1 + bA_1}{a - b} \\[1em] &= \dfrac{bA_1 - aA_2}{a - b} \end{align}$$

Here's a problem students can test the formula on:

They should get \(xy = 15\text{ m}^2.\)