Problem 1 - College Algebra with Modeling and Visualization

Question

Solve the rational equation (a) symbolically, (b) graphically, and (c) numerically $$ \frac{2 x}{x+2}=6 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -3\).

Step 1: Simplify the Equation Symbolically

Start by isolating the variable. Given: \( \frac{2x}{x+2} = 6 \). Multiply both sides by \(x + 2\) to clear the fraction: \(2x = 6(x + 2)\). Distribute the 6: \(2x = 6x + 12\). Rearrange the equation to gather like terms: \(2x - 6x = 12\), leading to \(-4x = 12\). Finally, divide by \(-4\): \(x = -3\).

Step 2: Graphical Solution

Graph both sides of the original equation. Consider \(y_1 = \frac{2x}{x+2}\) and \(y_2 = 6\). Plot these functions on the same coordinate plane. Find the x-coordinate where the graphs intersect. The intersection point confirms that \(x = -3\).

Step 3: Numerical Solution

Choose a value close to the expected solution to check: Substitute \(x = -3\) back into the equation \(\frac{2(-3)}{-3+2}\). Calculate: \(\frac{-6}{-1} = 6\), which is correct. This confirms \(x = -3\) as a solution numerically.

Key Concepts

Symbolic solution

Symbolic solutions are all about manipulating the equation to find the unknown variable. In this exercise, we have a rational equation \(\frac{2x}{x+2} = 6\). The goal is to solve for \(x\) using algebraic techniques while keeping the equation balanced.

Here's how it's done effectively:

• First, eliminate the fraction by multiplying both sides by the denominator, \(x + 2\). This gives us: \(2x = 6(x + 2)\).
• Next, distribute the \(6\) on the right side to get: \(2x = 6x + 12\).
• Then, gather all \(x\)-terms to one side of the equation by subtracting \(6x\) from both sides: \(2x - 6x = -4x = 12\).
• Lastly, solve for \(x\) by dividing both sides by \(-4\), resulting in \(x = -3\).

The main idea here is to "clear" the fraction and systematically work towards isolating \(x\). This step-by-step determination using algebraic rules is what makes it a symbolic solution.

Graphical solution

Graphical solutions provide a visual representation of solving equations. They can be very handy for confirming symbolic steps or when equations are tricky to solve algebraically.

For the equation \(\frac{2x}{x+2} = 6\), we interpret it graphically by plotting two functions:

• Function 1 (Left-Hand Side of the Equation): \(y_1 = \frac{2x}{x+2}\)
• Function 2 (Right-Hand Side of the Equation): \(y_2 = 6\)

Plot these on the same coordinate plane. The intersection point of the two graphs gives the value of \(x\) where both expressions are equal. In this example, the graphs intersect at \(x = -3\).

This visual approach helps to reaffirm the results obtained symbolically and provides an intuitive understanding of how solutions are derived in a real-world sense.

Numerical solution

A numerical solution is like testing a hypothesis by substituting numbers directly into the equation. The idea is to verify the solution found symbolically by plugging it back into the original rational equation.

In this exercise, we numerically test the result \(x = -3\). To do this, substitute \(x = -3\) back into the equation: \(\frac{2(-3)}{-3+2}\). The calculation simplifies to \(\frac{-6}{-1}\), which results in \(6\). This confirms that our symbolic solution \(x = -3\) is indeed correct numerically.

Numerical verification provides confidence that the algebraic manipulation was executed properly. It is a straightforward way to test the validity of the achieved solution, leaving no room for error in real-world applications or assessments.