Problem 97 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{-2 y+4}{4-5 y}-\frac{9}{5 y-4} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: $$ \frac{-2y+4}{4-5y} - \frac{9}{5y-4}. $$ Answer: The simplified expression is: $$ \frac{-10y^2+77y-124}{(4-5y)(5y-4)}. $$

Step 1: Find the Common Denominator

First, we need to find the common denominator by finding the least common multiple (LCM) of the denominators \((4-5y)\) and \((5y-4)\). Since both denominators are linear expressions and they are not multiples of each other, the LCM is their product: $$ \text{LCM}(4-5y,5y-4)=(4-5y)(5y-4). $$

Step 2: Adjust the Numerators

Now, we need to adjust each rational expression so that they have the same denominator, which is the LCM. We do this by multiplying the numerator and the denominator of each fraction by the missing factor: $$ \frac{-2y+4}{4-5y}\times\frac{5y-4}{5y-4}= \frac{(-2y+4)(5y-4)}{(4-5y)(5y-4)} $$ and $$ \frac{9}{5y-4}\times\frac{4-5y}{4-5y}= \frac{9(4-5y)}{(5y-4)(4-5y)}. $$

Step 3: Subtract the Expressions

Now that both fractions have the same denominator, we can subtract them: $$ \frac{(-2y+4)(5y-4)}{(4-5y)(5y-4)} - \frac{9(4-5y)}{(5y-4)(4-5y)}. $$ By subtracting the numerators and keeping the common denominator, we get: $$ \frac{(-2y+4)(5y-4)-9(4-5y)}{(4-5y)(5y-4)}. $$

Step 4: Simplify the Result

Now, let's simplify the numerator by performing the multiplications and combining like terms: $$ \frac{ (-10y^2+32y-16)-(108-45y)}{(4-5y)(5y-4)}. $$ Combine like terms in the numerator: $$ \frac{-10y^2+32y-16-108+45y}{(4-5y)(5y-4)}. $$ This simplifies further to: $$ \frac{-10y^2+77y-124}{(4-5y)(5y-4)}. $$ We cannot simplify this expression any further. So, the final answer is: $$ \frac{-10y^2+77y-124}{(4-5y)(5y-4)}. $$

Key Concepts

least common multiple

When dealing with rational expressions, finding a common denominator is crucial. This enables us to either add or subtract the fractions easily. To find a common denominator, we often use the least common multiple (LCM) of the denominators. In simpler terms, the LCM is the smallest expression that each denominator can divide into without leaving a remainder.
For linear expressions like the ones in the given problem \((4-5y)\) and \((5y-4)\), their LCM is just their product \((4-5y)(5y-4)\). This means that neither \((4-5y)\) nor \((5y-4)\) are factors of each other and we need their product for the common denominator.
This process guarantees that when each fraction is converted to have the LCM as its denominator, they can be confidently added or subtracted.

simplifying expressions

Simplifying expressions in mathematics means to rewrite them in their simplest form. This can involve combining like terms, reducing fractions, or factoring out common factors. In the context of the exercise, after adjusting each rational expression to have a common denominator, our task becomes simplifying the numerator.
This involves expanding the products \((-2y+4)(5y-4)\) and \(9(4-5y)\).Performing these multiplications: - First expand: \((-2y)(5y) + (-2y)(-4) + 4(5y) + 4(-4) \). - Then expand: \(9(4) + 9(-5y)\).
After performing these operations, you'll combine like terms from both expressions. This process helps reduce unnecessary complexity and presents the expression in a format that's easy to understand or solve in further steps.

subtracting fractions

Subtracting fractions, whether numerical or algebraic, follows a similar approach. You need a common denominator to carry out the operation correctly. Once we identified the least common multiple in the first step, we adjusted the numerators accordingly.
For our problem, after converting both fractions, we simply subtract the two expressions by focusing on their numerators. They now share a denominator, \((4-5y)(5y-4)\), making the subtraction straightforward.- Subtract: \[(-2y+4)(5y-4)\] from [9(4-5y)].
Remember, when dealing with polynomial numerators, each term from one numerator should be subtracted from each corresponding term in the other numerator. This operation, while simple in concept, involves keeping track of signs and combining like terms to simplify the final expression.
This is a consistent process that applies to any rational expression subtraction, and mastering it can be very helpful for mathematical success.