Question

Use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places. $$ \left\\{\begin{array}{l} y=x^{3 / 2} \\ y=e^{-x} \end{array}\right. $$

Short answer

The solutions are approximately \((0.64, 0.53)\) and \((2.19, 0.11)\).

Step-by-step solution

Identify the Equations

The system of equations is given as: 1. \( y = x^{3/2} \) 2. \( y = e^{-x} \)

Graph the First Equation

Plot the equation \( y = x^{3/2} \) using a graphing utility. This is a nonlinear equation that represents a curve.

Graph the Second Equation

Plot the equation \( y = e^{-x} \) using the same graphing utility. This is an exponential decay function.

Find the Intersection Points

Observe the graph and identify the points where the two curves intersect. These intersection points represent the solutions to the system of equations.

Round the Intersection Points

Round the coordinates of the intersection points to two decimal places to obtain the final solutions.

Key concepts

graphing utility

Graphing utilities are tools that help visualize mathematical functions and their intersections. These tools can be physical devices, like graphing calculators, or software applications available on computers and smartphones. By inputting equations into a graphing utility, one can plot curves and lines on a coordinate plane. This visualization makes it easier to understand the behavior of functions and identify solutions for systems of equations.
To solve the given system of equations, you can use a graphing utility to plot both functions. Follow these steps:

• Input the first equation, \( y = x^{3/2} \), into the graphing utility.
• Input the second equation, \( y = e^{-x} \), into the graphing utility.
• Observe the graph for points where the two curves intersect.
• These intersection points are the solutions to the system of equations.

Graphing utilities simplify the process of solving equations by providing a visual representation, making it easier to find precise intersection points.

exponential decay function

Exponential decay functions are mathematical models used to describe processes where quantities decrease rapidly at first and then level off over time. The general form of an exponential decay function is \( y = a e^{-bx} \), where:

• \(a\) is the initial amount or value.
• \(b\) is the decay constant.

In the given system, the function \( y = e^{-x} \) is an exponential decay function. Unlike linear functions, exponential decay functions decrease quickly for small values of \( x \) and then decrease more slowly as \( x \) increases.
This type of function is commonly used in various fields such as physics, biology, and finance to model phenomena like radioactive decay, population decline, and depreciation of assets.
By graphing \( y = e^{-x} \), you will see a steep drop that starts near the y-axis and approaches zero as \( x \) increases.

nonlinear equation

Nonlinear equations involve variables raised to a power other than one or involve other non-linear operations such as exponential, logarithmic, or trigonometric functions. In the given system, the equation \( y = x^{3/2} \) is a nonlinear equation. It is a function where the variable \( x \) is raised to a non-integer power (3/2). This results in a curve rather than a straight line when graphed.
The nature of nonlinear equations can make finding their solutions more challenging compared to linear equations. However, using a graphing utility to visualize the curve of \( y = x^{3/2} \) will help you see how it interacts with the exponential decay function \( y = e^{-x} \).
To recap, the main characteristics of nonlinear equations include:

• Graphs that form curves rather than straight lines.
• Solutions that may not be easily obtained through algebraic manipulation.
• Intersection points with other functions that indicate solutions in systems of equations.

Understanding nonlinear equations is essential for solving complex problems in various scientific and engineering disciplines.