Problem 54 - Introductory Algebra for College Students

Question

Solve or simplify, whichever is appropriate. $$\frac{4}{y-2}-\frac{1}{2-y}=\frac{25}{y+6}$$

Step-by-Step Solution

Answer

The solution to the equation is \(y=4\).

Step 1: Identify similar fractions

Observe that the denominators \(y-2\) and \(2-y\) of the two initial fractions on the left-hand side of the equation are opposite of each other. This gives us a clue that we can consider the first denominator \(y-2\) as \(-(2-y)\). So, \(\frac{4}{y-2}-\frac{1}{2-y}\) can be rewritten as \(\frac{4}{y-2}+\frac{1}{y-2}\).

Step 2: Combine the fractions

Now, combine the fractions on the left-hand side of the equation under a common denominator. This results in \(\frac{4+1}{y-2} = \frac{5}{y-2}\)

Step 3: Set up the equation

Now set the equation in the form \(\frac{5}{y-2} = \frac{25}{y+6}\)

Step 4: Cross-multiply and solve

Cross multiply the fractions to solve for \(y\). Doing so results in \(5(y+6) = 25(y-2)\). Expanding and simplifying the two sides of the equation leaves us with \(5y+30 = 25y-50\). Further simplifying the equation gives us \(20y = 80\). Solving for \(y\) we get \(y=4\).

Key Concepts

Equations

In algebra, equations are fundamental expressions that showcase the equality between two sides linked by the equal sign, '='. An equation typically involves variables, numbers, and operations. For instance, in the given problem, the equation is \[\frac{4}{y-2} - \frac{1}{2-y} = \frac{25}{y+6}\]This equation includes algebraic fractions, where the variable 'y' is in the denominator. To solve equations like this, our goal is to find the value of 'y' that satisfies the equality. Often, simplifying both sides of the equation or transforming it can help you find the solution. Here are some tips when dealing with equations:

• Always perform the same operation on both sides to maintain equality.
• Try to simplify where possible before solving for the variable.
• Look for patterns or transformations that can make the equation more straightforward, like adjusting denominators in fractions.

By understanding the structure of equations, you can approach solving them systematically. Remember, finding the variable's value might require several steps, including simplifying or transforming expressions.

Fractions

Fractions in algebra represent a division of terms and can sometimes complicate equations due to their denominators. The problem given involves fractions like \[ \frac{4}{y-2} \text{ and } \frac{1}{2-y} \]These fractions have denominators with variables, making them algebraic fractions. An essential first step in working with algebraic fractions is to look for ways to simplify them, such as combining them into a single fraction if possible.To add or subtract fractions:

• Find a common denominator.
• Rewrite each fraction using this common denominator.
• Combine the numerators while keeping the common denominator.

In our example exercise, we noticed that the denominators, \(y-2\) and \(2-y\), are opposites. Recognizing this allows us to rewrite the equation in a simpler form. This operation often leads to easier calculations and a clearer path to solving the problem.

Cross-Multiplication

Cross-multiplication is a useful method for solving equations that involve fractions set equal to each other, like in the equation\[ \frac{5}{y-2} = \frac{25}{y+6} \]This technique involves multiplying across the equal sign in a diagonal fashion. Specifically:

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Multiply the denominator of the first fraction by the numerator of the second fraction.
• Set the two products equal to each other.

For the equation above, the process is as follows:- Multiply 5 and \(y+6\)- Multiply 25 and \(y-2\)- Set the results from both operations equal: \[ 5(y+6) = 25(y-2) \]After performing these multiplications, you simplify the resulting equation to solve for the variable. Cross-multiplying can effectively remove the fractions from your equation, making it easier to isolate the variable and find the solution. Remember, the power of cross-multiplication lies in providing a straightforward pathway to simplify equations with fractions.