Problem 20 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{y-1}{y+2}=\frac{y+3}{y-2} $$

Step-by-Step Solution

Answer

Answer: The solution for the given rational equation is $y = -\frac{1}{2}$.

Step 1: Find a common denominator

Since the denominators of the fractions are \((y+2)\) and \((y-2)\), the common denominator of the two fractions is their product, \((y+2)(y-2)\).

Step 2: Multiply both sides of the equation by the common denominator

Multiply both sides of the equation by \((y+2)(y-2)\) to eliminate the fractions: $$ (y+2)(y-2)\left(\frac{y-1}{y+2}\right)=(y+2)(y-2)\left(\frac{y+3}{y-2}\right) $$ Now, divide both sides by the denominators to simplify the equation: $$ (y-1)(y-2) = (y+3)(y+2) $$

Step 3: Simplify the resulting equation and solve for y

Expand the expressions on both sides of the equation and then combine like terms: $$ y^2 - 2y - y + 2 = y^2 + 3y + 2y + 6 $$ $$ y^2 - 3y + 2 = y^2 + 5y + 6 $$ Now, subtract the left side of the equation from the right side to get all terms on one side: $$ 0 = 8y + 4 $$ Divide by 8 on both sides of the equation to solve for y: $$ y = -\frac{1}{2} $$ The solution to the given rational equation is \(y = -\frac{1}{2}\).

Key Concepts

Understanding Common Denominator

In rational equations, fractions often have different denominators, making them tricky to work with. One effective way to handle this is to find a **common denominator**. The common denominator of two fractions is essentially a shared multiple of their individual denominators. In our exercise, the denominators are \((y+2)\) and \((y-2)\). The simplest way to combine them is to multiply them together, resulting in the common denominator \((y+2)(y-2)\). This step is essential because it allows us to easily combine or eliminate fractions, turning the equation into a simpler form to solve. Remember, finding a common denominator is a vital step in making the equation simpler and more manageable.

Multiplying Both Sides of the Equation

Once a common denominator is identified, the next step is to **multiply both sides** of the equation by it. By doing this, we eliminate the fractions and are left with a polynomial equation. For our specific equation, multiplying both sides by \((y+2)(y-2)\) gives us:

• Left side: \((y+2)(y-2)\left(\frac{y-1}{y+2}\right)\)
• Right side: \((y+2)(y-2)\left(\frac{y+3}{y-2}\right)\)

This multiplication cancels out the denominators on each side, leaving us with a cleaner equation to solve. It's like clearing away the clutter to see the main structure of the problem.

How to Simplify the Equation

After multiplying both sides by the common denominator, the next goal is to **simplify the equation**. This involves first cancelling out the denominators, which we have already done, resulting in:

• \((y-1)(y-2) = (y+3)(y+2)\)

Next, we expand and combine like terms. Expanding means using the distributive property to multiply the individual terms in each binomial. For the left side:

• \(y^2 - 2y - y + 2\)

And for the right side:

• \(y^2 + 3y + 2y + 6\)

Combining like terms simplifies it further to \(y^2 - 3y + 2 = y^2 + 5y + 6\). The equation is now simpler and readier for the final step, solving it for \(y\).

Solving for y

Finally, with the equation simplified, it's time to **solve for \(y\)**. We need to bring all terms to one side of the equation so that we can solve it more easily. Subtract \(y^2 - 3y + 2\) from both sides to get \[0 = 8y + 4\].This rearrangement illustrates that our goal is to isolate \(y\). Now, divide each side by 8, which results in \[y = -\frac{1}{2}\].The division by 8 simplifies to a straightforward result, giving us the value for \(y\). The solution \(y = -\frac{1}{2}\) tells us the value at which our original rational equation holds true, effectively meaning both sides balance or are equal. Solving for \(y\) is the final step in freeing \(y\) from the equation format and finding its actual value.