Problem 36 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{2}{y+2}+\frac{8}{y+2}=\frac{9}{y+3} $$

Step-by-Step Solution

Answer

Answer: The value of y that satisfies the given rational equation is y = -12.

Step 1: Identify the common denominator

In our given equation, we have two denominators, y + 2 and y + 3. The common denominator will be the product of these two denominators: (y + 2)(y + 3).

Step 2: Clear the denominators

To clear the denominators in our rational equation, we will multiply each term on both sides of the equation by our common denominator, (y + 2)(y + 3). $$ ((y+2)(y+3))\left(\frac{2}{y+2}\right) + ((y+2)(y+3))\left(\frac{8}{y+2}\right) = ((y+2)(y+3))\left(\frac{9}{y+3}\right) $$

Step 3: Simplify the equation

Now we have to simplify the equation by canceling out the terms that appear in both the numerator and denominator. $$ (2(y+3)) + (8(y+3)) = 9(y+2) $$

Step 4: Distribute and combine like terms

Expand and combine like terms on each side. $$ 2y+6 + 8y+24 = 9y+18 $$ $$ 10y+30 = 9y+18 $$

Step 5: Solve for y

Isolate the variable y by subtracting 9y from both sides of the equation and subtracting 30 from both sides as well. $$ 10y-9y = 18-30 $$ $$ y = -12 $$

Step 6: Check for extraneous solutions

To check for extraneous solutions, we will plug the value of y back into the original equation and see if both sides are still equal. $$ \frac{2}{-12+2}+\frac{8}{-12+2}=\frac{9}{-12+3} $$ $$ \frac{2}{-10}+\frac{8}{-10}=\frac{9}{-9} $$ $$ \frac{-1}{5}+\frac{-4}{5}=\frac{-1}{1} $$ $$ \frac{-5}{5}=\frac{-5}{5} $$ The equation holds true for y = -12, so it is a valid solution. The solution for the given rational equation is: y = -12

Key Concepts

Common Denominator

When solving rational equations, finding a common denominator is a pivotal initial step. It's akin to preparing a solid foundation before building a house. In mathematical terms, a common denominator for a set of fractions is a shared multiple of their denominators. In the given exercise, the two denominators are \(y+2\) and \(y+3\). The least common denominator (LCD) is their product, \(y+2)(y+3)\). This helps combine fractions on the same side or between two sides of an equation, streamlining the process towards finding a solution.

Here's an analogy—imagine needing a uniform currency when shopping in a foreign country. Instead of using two types of currencies, you exchange them for a common one accepted by all vendors. Similarly, common denominators make it easier to work with and combine rational expressions, moving closer to a solution.

Clearing Denominators

The strategy of clearing denominators is about eliminating the fractions for a more manageable equation. By multiplying each term by the least common denominator, you effectively remove the denominational divide, just like paying off a debt completely to avoid dealing with interest rates. This process means that every term in the equation is now