Problem 78 - College Algebra with Modeling and Visualization

Question

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ 3(\pi-x)+\sqrt{2}=0 $$

Step-by-Step Solution

Answer

The solution to the equation is approximately 3.6.

Step 1: Isolate the Variable

Let's start with the given equation, which is:\[ 3(\pi - x) + \sqrt{2} = 0 \]Subtract \( \sqrt{2} \) from both sides:\[ 3(\pi - x) = -\sqrt{2} \]

Step 2: Solve for the Variable

Now, divide both sides by 3 to isolate \( \pi - x \):\[ \pi - x = -\frac{\sqrt{2}}{3} \]Next, solve for \( x \) by subtracting \( \pi \) from both sides:\[ -x = -\frac{\sqrt{2}}{3} - \pi \]Multiply the entire equation by -1 to solve for positive \( x \):\[ x = \pi + \frac{\sqrt{2}}{3} \]

Step 3: Compute the Numerical Solution

Using a calculator, approximate the solution by substituting the values:\[ x \approx 3.14159 + \frac{1.41421}{3} \]Calculate \( \frac{1.41421}{3} \approx 0.4714 \), so:\[ x \approx 3.14159 + 0.4714 = 3.613 \]Rounded to the nearest tenth, \( x \approx 3.6 \).

Step 4: Graphical Verification

Graph the functions \( y = 3(\pi - x) + \sqrt{2} \) and \( y = 0 \) to find the intersection point, which represents the solution. The x-coordinate of the intersection, as calculated, should be approximately 3.6.

Key Concepts

Symbolic Solutions

Solving equations symbolically involves manipulating algebraic expressions to find an exact solution. In our problem, we have the equation \(3(\pi - x) + \sqrt{2} = 0\). The goal is to isolate \(x\):

• First, remove constants from one side by subtracting \(\sqrt{2}\), giving us \(3(\pi - x) = -\sqrt{2}\).
• Next, divide each side by 3 to get \(\pi - x = -\frac{\sqrt{2}}{3}\). This simplification helps to bring \(x\) out.
• Rearrange to solve for \(x\): \(-x = -\frac{\sqrt{2}}{3} - \pi\).
• By multiplying through by -1, you get \(x = \pi + \frac{\sqrt{2}}{3}\).

This final expression for \(x\) provides the symbolic solution, which is an exact answer written in terms of \(\pi\) and \(\sqrt{2}\). Symbolic solutions are precise and do not rely on approximations or estimations.

Graphical Solutions

Graphical solutions involve plotting the equation to visualize where it crosses the x-axis or intersects with another graph. For the equation \(3(\pi - x) + \sqrt{2} = 0\), you can analyze it graphically by defining functions:

• Define \(y_1 = 3(\pi - x) + \sqrt{2}\) and \(y_2 = 0\).
• Plot these functions on a graph. In this plot, \(y_2 = 0\) is the x-axis itself.
• The intersection of \(y_1\) with the x-axis \((y_2)\) corresponds to the value of \(x\) that solves the equation.

Using a graphing calculator or software, plot these functions to pinpoint where \(y_1\) crosses the x-axis. In this exercise, the intersection at approximately \(x = 3.6\) visually confirms the solution derived symbolically and numerically. Graphical methods provide a powerful way to see the behavior of equations, especially when solutions can be approximations or when understanding the relationship between variables is key.

Numerical Solutions

Numerical solutions involve using approximations and calculations to find an approximate value for \(x\). In this exercise, we already have a symbolic expression for \(x\) as \(x = \pi + \frac{\sqrt{2}}{3}\).

• Start by substituting known approximations like \(\pi \approx 3.14159\) and \(\sqrt{2} \approx 1.41421\).
• Calculate \(\frac{1.41421}{3} \approx 0.4714\).
• Add this result to \(3.14159\) to obtain \(x \approx 3.14159 + 0.4714 = 3.613\).
• Finally, round to the nearest tenth to get \(x \approx 3.6\).

Numerical solutions are especially useful when exact symbolic solutions are difficult to compute or when a practical approximation suffices. They provide a seamless bridge between theory and real-world applications, ensuring you can always come down to exact or practical number representations for theoretical expressions.