Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the given equation. $$ \frac{8}{x}=16 $$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{8}{x} = 16\) is \(x = \frac{1}{2}\).

Step 1: Understand the given equation

In this case, we have a fraction \(\frac{8}{x}\) equal to 16.

Step 2: Isolate the variable x on one side of the equation

We want to get rid of the denominator, so we can multiply both sides of the equation by \(x\). $$ x \cdot \frac{8}{x} = 16 \cdot x \\ 8 = 16x $$

Step 3: Solve for x

Now we just need to divide both sides by 16 to get x by itself. $$ \frac{8}{16} = \frac{16x}{16} \\ \frac{1}{2} = x $$

Step 4: Check the solution

Finally, let's check our solution by plugging the value of \(x\) back into the original equation: $$ \frac{8}{\frac{1}{2}} = 16 $$ Since \(8 \div \frac{1}{2} = 8 \cdot 2 = 16\), our solution is correct.

Step 5: Final Answer

The solution to the equation \(\frac{8}{x} = 16\) is \(x = \frac{1}{2}\).

Key Concepts

Solving Equations

Solving equations is an essential skill in mathematics. It consists of finding the value of the variable that makes an equation true. Each equation typically balances two expressions against each other, connected by an equal sign. Our goal is to find which value(s) will make both sides equal.
In this exercise, we started with the equation \(\frac{8}{x} = 16\). Here, 8 divided by some unknown number \(x\) equals 16. Solving equations often requires manipulating these expressions using arithmetic operations to unveil the unknown value.
Key steps in solving equations usually include:

• Understanding what each term represents.
• Performing operations to simplify the equation logically.
• Checking if the solution satisfies the original equation.

This methodical approach ensures you find the correct value for the variable and that your solution is valid.

Fractional Equations

Fractional equations are those that involve fractions, meaning one or more terms are expressed with a numerator and a denominator. In our exercise, the fractional term is \(\frac{8}{x}\).
When solving fractional equations, it's useful to eliminate fractions to simplify the process. This is often done by multiplying both sides by the denominator, which in turn "cancels out" the fraction.
In the equation \(\frac{8}{x} = 16\), we isolated the fraction by multiplying the entire equation by \(x\). Doing this transformed the equation into \(8 = 16x\), a simpler linear form. This makes it easier to see how to solve for \(x\).

• Fractional equations benefit from clearing fractions early on.
• Keep the equation balanced by performing the same operation on both sides.
• Be cautious of any restrictions on variable values (e.g., division by zero).

Mastering these techniques allows you to solve any fractional equation confidently.

Isolating Variables

Isolating variables means rearranging an equation so that the variable you're solving for is alone on one side. This step is crucial for finding a solution. It typically involves arithmetic operations like addition, subtraction, multiplication, or division.
After we multiplied the equation by \(x\), we needed \(x\) by itself. This led us to the equation \(8 = 16x\). By dividing both sides by 16, we isolated \(x\) to find it equaled \(\frac{1}{2}\).
Here are some helpful pointers:

• Focus on simplifying the equation until the variable stands alone.
• Use inverse operations (e.g., multiplication and division are inverses) to move components across the equal sign.
• Always maintain equation balance to ensure it remains true.

These strategies are central in algebra and help achieve the goal of isolating the desired variable, simplifying the problem-solving process.