Problem 40 - Elementary and Intermediate Algebra

Question

Solve each equation and check the result. If an equation has no solution, so indicate. $$ \frac{3}{y-2}=\frac{3}{y-2}+1 $$

Step-by-Step Solution

Answer

The equation has no solution.

Step 1: Identify the Equation

We are given the equation \( \frac{3}{y-2} = \frac{3}{y-2} + 1 \). Our first task is to recognize the equation set up as a rational equation, which involves fractions.

Step 2: Subtract the Same Term

Notice that \( \frac{3}{y-2} \) appears on both sides of the equation. Subtract \( \frac{3}{y-2} \) from both sides to simplify, yielding \( 0 = 1 \).

Step 3: Analyze the Result

The equation \( 0 = 1 \) is a contradiction, meaning the original equation asked us to find a value of \( y \) that makes this true, which is impossible.

Step 4: Conclusion

Since we derived a contradiction, the original equation \( \frac{3}{y-2} = \frac{3}{y-2} + 1 \) has no solution. This means no value of \( y \) will satisfy the equation.

Key Concepts

Rational Equations

Rational equations play an important role in algebra. They typically involve fractions where the variables appear in the denominators. For instance, in the equation \( \frac{3}{y-2} = \frac{3}{y-2} + 1 \), the fractions are \( \frac{3}{y-2} \).
To solve such equations, it's often useful to first simplify the fractions or clear the fractions by finding a common denominator. However, in this case, you would notice something interesting: the same term, \( \frac{3}{y-2} \), appears on both sides.
Rational equations can sometimes be tricky because you need to be mindful of values that can make the denominator zero. These will result in undefined expressions, which will affect the solution set. Always check for these values, as they may need to be excluded from your final answer.

Solving Equations

When solving equations, your goal is to isolate the variable and determine the value(s) that make the equation true. For the equation \( \frac{3}{y-2} = \frac{3}{y-2} + 1 \), a clear first step is to subtract \( \frac{3}{y-2} \) from both sides. This is done to cancel out the fractions and simplify the equation.
Upon doing this, you’re left with \( 0 = 1 \). Here, something unusual happens. You end up with a statement that's clearly false because zero never equals one. This indicates that, instead of finding a value for \( y \), the equation is set up such that there is no possible value of \( y \) that makes it true.

No Solution

In algebra, finding a solution means identifying a value for the variable that satisfies the equation. However, not all equations have solutions. When an equation reduces to a false statement such as \( 0 = 1 \), it means it has no solution.
This particular equation reveals that no matter what value \( y \) takes, the two sides of the equation will never be equal. Therefore, the equation \( \frac{3}{y-2} = \frac{3}{y-2} + 1 \) cannot be satisfied by any \( y \).
Such situations often occur when terms cancel out completely, leaving behind a contradiction. It's a useful reminder that sometimes equations are structured in a way that inherently defies a solution. In practice, this means you'd state clearly that the equation has no solution, rather than attempting to find a value that does not exist.