Problem 65 - Contemporary Precalculus

Question

Solve the equation and check your answers. $$\frac{1}{2 t}-\frac{2}{5 t}=\frac{1}{10 t}-1$$

Step-by-Step Solution

Answer

Answer: There is no valid solution for the given equation, as the apparent solution $$t=0$$ results in undefined fractions and a division by 0.

Step 1: Find the common denominator

To eliminate the fractions, we need to find the lowest common denominator (LCD) of the denominators $$2t, 5t,$$ and $$10t$$. The LCD is the smallest multiple of these denominators. In this case, the LCD is $$10t$$.

Step 2: Multiply each term by the LCD

Now, multiply each term in the equation by the LCD and simplify. $$10t \times \left(\frac{1}{2 t}-\frac{2}{5 t}=\frac{1}{10 t}-1\right)$$ $$\Rightarrow 5-4=1-10t$$

Step 3: Solve for t

Now, we only have one variable to solve for, and the equation is simple. We will isolate t by adding $$10t$$ to both sides and then dividing by ten. $$5-4+10t=1-10t+10t$$ $$1+10t=1$$ Subtract 1 from both sides: $$10t=0$$ Divide by 10: $$t=0$$

Step 4: Check the solution

To check our solution, we will plug t back into the original equation and see if it holds true. $$\frac{1}{2 (0)}-\frac{2}{5 (0)}=\frac{1}{10 (0)}-1$$ However, the solution $$t=0$$ causes all the denominators to be 0, which results in undefined fractions and a division by 0. Therefore, this equation has no valid solution.

Key Concepts

Common Denominator

When solving equations with fractions, finding the common denominator is an essential first step. A common denominator is a shared multiple of the denominators of the fractions involved. For example, in the equation \( \frac{1}{2t} - \frac{2}{5t} = \frac{1}{10t} - 1 \), the denominators are \(2t\), \(5t\), and \(10t\). The least common multiple of these denominators is the smallest number that all three can divide into evenly, which is \(10t\) in this case.

• To find the common denominator, identify the largest of the given denominators and determine if the other denominators can divide into it.
• If not, calculate multiples of the largest denominator until you find one that all can share.
• This step allows you to eliminate the fractions by equalizing all terms, simplifying the equation to a format where solving is more straightforward.

Once the common denominator is identified, multiply through the equation by this value to cancel out the fractions, smoothing the path towards solving the equation.

Solving Equations

After eliminating fractions by using a common denominator, the next step is solving the resulting equation. Let's take the simplified version derived from the previous example: \(5 - 4 = 1 - 10t\). This equation may seem simple, but careful steps must be taken to isolate and solve for the variable \(t\).
First, aim to collect all terms containing the unknown variable on one side of the equation and constants on the other side. In our example:

• Add \(10t\) to both sides to remove it from the right side: \(5 - 4 + 10t = 1\).
• After simplifying, we have \(1 + 10t = 1\).

Subtract \(1\) from both sides, yielding \(10t = 0\). Finally, divide by \(10\), resulting in \(t = 0\). Remember, solving an equation is about maintaining balance, adjusting both sides equally with each arithmetic operation.

Undefined Fractions

An undefined fraction occurs when its denominator equals zero. Mathematically, division by zero is undefined because it disrupts arithmetic laws, breaking the balance required for equations. If you encounter a scenario like this when solving equations, it is critical to recognize it and adjust your approach.
Looking back at our exercise, when we obtained \(t = 0\) and verified it in the original equation, every fraction’s denominator became zero, which leads to undefined expressions. As soon as such a situation is detected while checking a solution, it invalidates the result, and the initial assumption, such as \(t = 0\), cannot provide a valid solution.

• Always substitute your solution back into the original equation to ensure none of the denominators become zero.
• If they do, rethink the solution methodology, considering restrictions that the variable may have.

Recognition of undefined fractions is crucial to accurately solving and validating equations.

Division by Zero

Division by zero is a concept in mathematics that leads to undefined results and is considered impossible to perform. When a denominator equals zero in any arithmetic operation or equation, the result cannot be determined within the realm of real numbers.
In our exercise, substituting \(t = 0\) into any fraction results in an undefined expression because it involves division by zero. This fundamental rule dictates that the equation has no solution because the necessary condition for defining any fraction is violated.

• Whenever solving equations, always check for values that could result in division by zero.
• If such a case arises, adjust your solution or state that no solution exists given the constraints.

Understanding this concept ensures you avoid logical errors, maintaining accuracy and validity while solving mathematical problems.