Problem 33 - Algebra and Trigonometry

Question

\(31-42\) . Solve the equation both algebraically and graphically. $$ \frac{2}{x}+\frac{1}{2 x}=7 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{5}{14} \).

Step 1: Rewrite the Equation

The given equation is \( \frac{2}{x} + \frac{1}{2x} = 7 \). First, combine the terms on the left-hand side to have a single denominator. This gives \( \frac{4}{2x} + \frac{1}{2x} = 7 \). Thus, we can rewrite the equation as \( \frac{5}{2x} = 7 \).

Step 2: Solve for x Algebraically

Rewrite \( \frac{5}{2x} = 7 \) by multiplying both sides by \( 2x \) to eliminate the fraction. This results in \( 5 = 14x \). Solving for \( x \), we divide both sides by 14: \( x = \frac{5}{14} \).

Step 3: Graph Both Sides

To solve graphically, we can plot the left side \( \frac{2}{x} + \frac{1}{2x} \) as one function and \( 7 \) as another. Plot \( y = \frac{5}{2x} \) and \( y = 7 \) on the same graph. The solution to the equation occurs where these two curves intersect.

Step 4: Identify the Intersection Point

The intersection of the two graphs occurs at the point where \( x = \frac{5}{14} \). This confirms our algebraic solution as the x-value of the intersection point is the same, showing the consistency of both methods.

Key Concepts

Graphical Solution

In mathematics, using graphical solutions to solve equations is a visual method. It involves plotting the expressions on either side of the equation as two separate graphs. For the given equation, where the expression is \( \frac{2}{x} + \frac{1}{2x} = 7 \), we'd handle this by plotting both:

• \( y = \frac{5}{2x} \)
• \( y = 7 \)

The essence of this process is to find the point at which these two graphs intersect. This intersection signifies the solution of the equation. When two graphs meet, it means their expressions are equal at that specific x-value. Here, plotting these on a coordinate plane, we find they intersect at \( x = \frac{5}{14} \). This point confirms that the x-value satisfies the equation. Graphical solutions are especially helpful for visual learners and provide a different perspective on understanding solutions beyond numbers and symbols.

Algebraic Solution

An algebraic solution involves manipulating the equation to find the unknown variable, using known algebraic techniques. First, let's consider the equation \( \frac{2}{x} + \frac{1}{2x} = 7 \). In solving this algebraically, the primary goal is to simplify the left-hand side by combining terms. This results in \( \frac{5}{2x} = 7 \). To eliminate the fraction, multiply both sides by \( 2x \). This removes the denominator and yields the equation \( 5 = 14x \). The next step involves isolating \( x \) to find its value by dividing both sides by 14:

• \( x = \frac{5}{14} \)

This process shows the necessity of consistently isolating the variable to uncover its value. Additionally, algebraic solutions allow for exact precision, which can sometimes be preferable over approximate or rounded solutions that result from graphical methods.

Solving Equations

Solving equations is a fundamental skill in algebra, comprising various methods including algebraic manipulation and graphical representation. The process involves identifying the value of the unknown variable that satisfies the equation. There are essential steps and strategies one often follows when solving equations, such as:

• Simplifying expressions and terms to make calculations straightforward.
• Using inverse operations (like addition, subtraction, multiplication, and division) to isolate the variable.
• Verifying with multiple methods, like graphical solutions, to confirm accuracy.

For the equation \( \frac{2}{x} + \frac{1}{2x} = 7 \), the steps we used are a classic example. Simplification and isolation revealed \( x = \frac{5}{14} \), while verification with a graph manifested in the intersection point of the two functions. Understanding these methods equips students with the tools necessary for tackling a wide range of mathematical equations across various complexities.