Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to solve the equation. (Round your solution to three decimal places.) \(\frac{2}{7.398}-\frac{4.405}{x}=\frac{1}{x}\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 19.945\). This is rounded to three decimal places.

Step 1: Dealing with Fractions

To make the equation easier to handle, first multiply the entire equation by \(7.398*x\). This will clear the fractions. This will result in: \(2x - 4.405*7.398 = 7.398\)

Step 2: Simplifying and Rearranging the Equation

Simplify the equation to see the problem more clearly. \(2x - 32.49229 = 7.398\). Next,arrange the terms in the equation to isolate x on one side,yielding: \(2x = 7.398 + 32.49229\)

Step 3: Solve for x

Now, solve the equation for x, \(2x = 39.89029\). Therefore, \(x = 39.89029 / 2\)

Key Concepts

Fractions in Equations

When solving algebraic equations, you might often encounter fractions. Fractions in equations can make the process appear daunting because they involve division. However, understanding what's happening beneath the surface in fractions can demystify these mathematical expressions.

• Fractions are essentially division problems showing how much you have (the numerator) out of the total parts (the denominator).
• In equations, fractions can represent parts of a whole or rates.
• They allow precise division and pairing of quantities, making them helpful in various scenarios.

It's common to see fractions in equations involving rates, ratios, or proportions. In the given example, each fraction aligns specific values to be compared and balanced. Becoming comfortable with fractions can make solving equations much more straightforward.

Clearing Fractions

Clearing fractions is a strategic step in simplifying equations, making them easier to solve. In equations with fractions, all terms are multiplied by a number that cancels out the denominators.

• This step eliminates the division inherent in fractions, simplifying the equation.
• It results in an equation without fractions, making it easier to solve (especially for beginners).
• Often, the least common denominator (LCD) or a multiplication of all denominators is used to clear the fractions.

In the provided solution, multiplying both sides of the equation by\(7.398 \times x\) effectively clears the fractions. This action turns a complex equation into a linear equation, paving the way for further solution steps. You remove the fraction from being a barrier to understanding, revealing a more direct relation between the variables.

Isolating Variables

Isolating variables is a key step towards solving equations. It involves rearranging the equation to get the variable you're solving for by itself on one side. This transformation helps in understanding what the variable "x" equals in terms of other numbers or variables.

• First, consolidate all terms involving the variable on one side (typically the left).
• Move all constants and terms unrelated to the variable to the opposite side.
• Use addition, subtraction, multiplication, or division to simplify as necessary.

For the solution to our exercise, the process involves moving terms around to isolate the variable "x". This is done by adding or subtracting terms from both sides until you have a clear expression for "x". For example, you rearrange \(2x = 7.398 + 32.49229\) so that only "x" is left on one side, and you solve for it by performing simple arithmetic operations. Mastering this technique is crucial for tackling more complex algebraic equations successfully.