Problem 30 - Contemporary Precalculus

Question

Use algebraic, graphical, or numerical methods to find all real solutions of the equation, approximating when necessary. $$\frac{2 x}{x+5}=1$$

Step-by-Step Solution

Answer

Question: Solve the equation \(\frac{2 x}{x+5}=1\) for all real solutions. Answer: x = 5

Step 1: Remove the fraction

To remove the fraction, we can simply multiply both sides of the equation by the denominator \((x+5)\), which will result in: $$(x+5)\left(\frac{2 x}{x+5}\right)=(x+5)(1)$$ Simplifying, we get: $$2x = x + 5$$

Step 2: Solve for x

Now we need to solve the equation for x by isolating it on one side. First, let's subtract x from both sides: $$2x - x = x - x + 5$$ Which simplifies to: $$x = 5$$

Step 3: Verify the solution

Finally, let's verify our solution by substituting it back into the original equation and making sure both sides are equal: $$\frac{2(5)}{5+5}=\frac{10}{10}=1$$ Indeed, both sides of the equation are equal, so our solution \(x=5\) is correct. The real solution to the equation \(\frac{2 x}{x+5}=1\) is \(x=5\).

Key Concepts

Algebraic Methods

Algebraic methods are a standard approach for solving equations and often the most straightforward. They involve manipulating the equation using operations such as addition, subtraction, multiplication, or division.
These methods aim to isolate the variable on one side to find its value. In our exercise, the equation \(\frac{2x}{x+5}=1\) requires this method.

• The first step is to eliminate any fractions, which simplifies operations.
• Next, rearrange the resulting expressions to solve for the variable.

Using algebraic methods helps reveal exactly where the solution comes from, giving us precise results and learning opportunities.

Graphical Methods

Graphical methods involve visual representations of equations. This approach is helpful when you want to see where two expressions meet or understand the behavior of equations.
To solve the equation \(\frac{2x}{x+5}=1\) graphically:

• Graph the left-hand side and right-hand side of the equation separately.
• The intersection point(s) on the graph will represent the real solutions.

Even though we found \(x=5\) algebraically, plotting these can confirm the solution or show additional insights if the equation were more complex.

Numerical Methods

When algebraic or graphical methods become cumbersome, numerical methods are an excellent alternative. They are approaches that approximate solutions using iterations and algorithms.
The most common numerical methods include:

• Bisection Method: This narrows down the interval where a solution exists.
• Newton's Method: This uses tangents to converge rapidly to a solution from an initial guess.

For our equation \(\frac{2x}{x+5}=1\), numerical solutions may not be necessary because it’s quite straightforward. However, for more complex situations, numerical methods are invaluable.

Real Solutions

Real solutions refer to the values of the variable that satisfy the equation and are real numbers—not imaginary. In the context of our exercise, we are seeking values of \(x\) that make the equation true.

• After solving \(\frac{2x}{x+5}=1\), we found that \(x=5\) is the only real solution.
• Verification is a crucial step; it confirms that substituting back satisfies the original equation.

For other equations, especially polynomials of higher degrees, there might be multiple real solutions, each representing a point where the equation is satisfied.

Equation Solving Steps

Solving equations typically follows a structured sequence of steps. This framework helps ensure no step is missed and the solution is accurate.
For our equation \(\frac{2x}{x+5}=1\), these steps are essential:

• Clear any fractions: Multiply both sides by the denominator to eliminate fractions.
• Simplify and isolate the variable: Rearrange the terms to get the variable alone.
• Verify: Substitute the solution back into the original equation to ensure it's valid.

Following these steps not only gives the solution but also builds a strong understanding of the principles of equation solving.