Problem 33 - Precalculus Mathematics for Calculus

Question

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: Simplify the Equation

Combine the terms on the left side of the equation. Both terms have a common denominator of \( x \). Thus, rewrite the equation as follows:\[\frac{2}{x} + \frac{1}{2x} = \frac{4}{2x} + \frac{1}{2x} = \frac{5}{2x}.\] Now the equation becomes \( \frac{5}{2x} = 7 \).

Step 2: Eliminate the Fraction

To eliminate the fraction, multiply both sides of the equation by \( 2x \) to obtain:\[ 5 = 7(2x). \]

Step 3: Solve for x

Divide both sides by 14 to solve for \( x \):\[ x = \frac{5}{14}. \]

Step 4: Graph the Functions

To solve graphically, plot the functions \( y = \frac{5}{2x} \) and \( y = 7 \) on the same graph. The solution to the equation is the x-coordinate of the point where these two graphs intersect.

Step 5: Identify Intersection Point

Check from the graph that the intersection point occurs at \( x = \frac{5}{14} \). Thus, the graphical solution matches the algebraic solution.

Key Concepts

Solving Equations

Solving algebraic equations can sometimes feel tricky, but with practice, it becomes easier. The fundamental aim is to find the value of the variable that makes the equation true. Let's look into the equation from the exercise:
\[ \frac{2}{x} + \frac{1}{2x} = 7 \] First, it's important to simplify the equation. Combine the fractions by using a common denominator, which in this case is the variable \(x\). This gives us \(\frac{5}{2x} = 7\).
The next step is to eliminate the fraction. This is done by multiplying both sides by \(2x\), resulting in:
\[ 5 = 7(2x) \]Now, we need to isolate \(x\). Divide both sides by 14:
\[ x = \frac{5}{14} \] This value of \(x\) is the solution to the equation. Remember, each step in solving the equation is like hiking up a mountain trail; take it one step at a time, and you'll reach the summit with success!

Graphical Representation

Visualizing equations through graphs can offer a powerful way to understand solutions. By plotting the related functions, you can see the solution as the point where two graphs meet. Here's how you apply this concept to our problem:
You'll need to plot two functions:

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

The graph of \( y = 7 \) is a horizontal line, as it shows that for any \(x\), \(y\) remains 7. Meanwhile, \( y = \frac{5}{2x} \) is a hyperbola, showing an inverse relationship between \(x\) and \(y\).
The solution to the equation is found at the intersection of these graphs. If you imagine drawing the graphs on the same axes, the x-coordinate of the point where they meet is \( x = \frac{5}{14} \).
Graphical representation not only helps confirm the solution but also deepens your understanding by providing a visual image of how solutions relate to graphs.

Algebraic Manipulation

Algebraic manipulation is like the toolkit of solving equations. It involves using various algebraic techniques to simplify and solve equations, helping you to better understand relationships between variables.
For example, in the given problem:

• Start by combining like terms to simplify the left side of the initial equation, finding a common denominator.
• Next, eliminate fractions by clearing denominators, making expressions easier to work with.
• Then, isolate the variable by dividing or multiplying both sides by necessary terms.

Consider how in our exercise, we combined fractions using a common denominator to simplify (\(\frac{5}{2x} = 7\)). Then, we multiplied through by \(2x\) to tackle the denominator head-on, resulting in \(5 = 7(2x)\). Finally, from here, a simple division gives us the solution \( x = \frac{5}{14} \).
By mastering algebraic manipulation, you can handle equations with confidence, knowing the steps to take that will lead you to the solution.