Problem 11 - Algebra for College Students

Question

Find the solution set to each equation. $$\frac{3}{x-2}+\frac{5}{x}=\frac{10}{x}$$

Step-by-Step Solution

Answer

The equation has no solutions.

Step 1: Identify the given equation

The given equation is \[\frac{3}{x-2} + \frac{5}{x} = \frac{10}{x}\].

Step 2: Combine like terms on the right side

Combine the fractions on the right side by subtracting \[\frac{5}{x}\] from \[\frac{10}{x}\], resulting in \[\frac{10}{x} - \frac{5}{x} = \frac{5}{x}\]. Therefore, the equation becomes: \[\frac{3}{x-2} + \frac{5}{x} = \frac{5}{x}\].

Step 3: Subtract \(\frac{5}{x}\) from both sides

Subtract \[\frac{5}{x}\] from both sides of the equation to isolate the fraction involving \[\frac{3}{x-2}\]. This results in: \[\frac{3}{x-2} = 0\].

Step 4: Solve for x

Since \[\frac{3}{x-2}\] equals zero, the numerator must be zero. Therefore, 3 is nonzero, indicating that \[\frac{3}{x-2}\] cannot be zero. Hence, there are no solutions for this equation.

Key Concepts

rational equations

Rational equations are equations that involve fractions with polynomials in the numerator, denominator, or both. These types of equations can seem a bit tricky at first, but they are quite manageable if you follow a clear process. In rational equations, the unknown variable appears in the denominator, which adds a layer of complexity. For instance, in our original equation \(\frac{3}{x-2} + \frac{5}{x} = \frac{10}{x}\), both fractions have variables in their denominators. To solve these equations, we often need to find a common denominator, clear the fractions, combine like terms, and isolate the variable.

isolate variables in equations

Isolating the variable means getting the variable by itself on one side of the equation. This step is crucial as it helps to directly find the value of the variable. The initial step might involve simplifying the equation. Consider our equation again: \(\frac{3}{x-2} + \frac{5}{x} = \frac{10}{x}\). First, we combine the fractions on the right side, \(\frac{10}{x} - \frac{5}{x} = \frac{5}{x}\). Next, to isolate the variable, we subtract \(\frac{5}{x}\) from both sides, reducing our equation to \(\frac{3}{x-2} = 0\). This process simplifies the equation step by step and makes isolating the variable much easier.

no solution equations

Not all equations have solutions. This is especially true for some rational equations. When we reach a step where our equation simplifies to a statement that cannot be true, we have a no solution scenario. For example, when we simplified our given equation to \(\frac{3}{x-2} = 0\), we noticed that it is not possible for a non-zero number (3) to be divided by any number and result in zero. Thus, the equation has no solutions. This type of insight is crucial. Always check the validity of the simplified expressions to determine if a solution exists.