Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{aligned} 2 \ln x+y &=4 \\ e^{x}-y &=0 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution can be found by identifying where the graphs of the functions \(y = 4 - 2\ln{x}\) and \(y = e^x\) intersect. Those intersection points provide the x and y values that solve the system of equations.

Step 1: Identify the Equations

Identify the two equations in the system. The given system of equations is: \[2 \ln{x} + y = 4\] and \[e^{x} - y = 0 \] or \[y = e^x\]. These will be the equations to plot.

Step 2: Plotting the First Equation

Begin by isolating y in the first equation to get the function to plot. This simplifies tracing the function on the graph. The first equation \(2\ln{x} + y = 4\) can be rewritten as \(y = 4 - 2\ln{x}\). Plot this function.

Step 3: Plotting the Second Equation

The second equation is already isolated so it can be plotted directly. The second equation is literally \(y = e^x\). Plot this function on the same graph.

Step 4: Find Intersection Points

Intersection points of two functions are solutions of the system. That's because at these points the x and y values satisfy both equations simultaneously. So, find the points where the two graphs from steps 2 and 3 intersect.

Key Concepts

Graphical Method

The graphical method is a visual approach to solving systems of equations by representing each equation as a graph. To solve a system graphically, you plot each equation on the same set of axes and identify the point(s) where the graphs intersect. These intersection points represent the solutions to the system, as they satisfy all equations simultaneously.

Using this method, you gain a clear, visual understanding of how different functions interact. To apply the graphical method effectively, follow these steps:

• Convert each equation into a function form, if necessary. For example, the equation \(2\ln{x} + y = 4\) is rearranged to \(y = 4 - 2\ln{x}\).
• Plot each function on a coordinate graph. In our example, plot \(y = 4 - 2\ln{x}\) and \(y = e^x\).
• Observe where the graphs intersect, as these points are the solutions.

The graphical method is straightforward for simple systems, but it may not be the most precise for complex systems or when exact solutions are required.

Algebraic Method

The algebraic method involves solving systems of equations using algebraic techniques, such as substitution or elimination, without graphing. This method is particularly useful for finding exact solutions numerically.

Let's break down two common algebraic approaches:

• Substitution Method: Solve one equation for one variable and substitute this expression into the other equation. For instance, begin with \(y = e^x\) and substitute into \(2\ln{x} + y = 4\), resulting in \(2\ln{x} + e^x = 4\). You can then solve for \(x\).
• Elimination Method: This method is best when you can eliminate one variable by adding or subtracting the equations. However, in this exercise, substitution is more practical.

The algebraic method provides precise results and is very effective when graphical solutions are impractical. It becomes complex with non-linear systems but offers a more exact approach to finding the solutions.

Exponential Functions

Exponential functions are mathematical functions of the form \(f(x) = a \cdot e^{bx}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828. These functions are characterized by rapid growth and are frequently used in modeling real-world phenomena, such as population growth or radioactive decay.

In the context of the given exercise, we deal with the function \(y = e^x\). Key features of exponential functions include:

• They grow (or decay) exponentially, meaning their rate of change increases (or decreases) as \(x\) increases.
• The graph of an exponential function rises sharply after a certain point and is always above the x-axis for positive values of \(a\).

Exponential functions are one of the fundamental function types in mathematics and have distinct uses in sciences and engineering.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and can be represented as \(f(x) = \log_b{x}\), where \(b\) is the base of the logarithm. In cases where the natural logarithm is used, the base is \(e\), written as \(\ln{x}\).

These functions are essential in many areas of math due to their unique properties:

• They compress rapidly increasing ranges of output into manageable scales, making them useful in diverse fields like acoustics and earthquake measurement (Richter scale).
• The graph of \(\ln{x}\) is concave down, passing through the point \((1,0)\), with a vertical asymptote at \(x = 0\).

In the exercise explored, the logarithmic equation \(2\ln{x} + y = 4\) represents one of the system's components, highlighting the connectivity between exponential and logarithmic functions.