Problem 47 - Precalculus with Limits

Question

Solving a System of Equations Graphically In Exercises \(45-48,\) use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places. $$\left\\{\begin{aligned} y+2 &=\ln (x-1) \\ 3 y+2 x &=9 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The exact solution will depend on the graphical approach used and may require a computational tool or graphing calculator to find the intersection point(s) accurately. Make sure to report your answer up to two decimal places.

Step 1: Rearrange the Equations

Rewrite the equations in a form that will be easy to graph. The first equation can be rewritten as \(y = \ln(x - 1) - 2\). The second equation can be manipulated as follows: \(3y = 9 - 2x\), thus \(y = (9 - 2x) / 3\). This forms a line equation \(y = mx + c\), where \(m\) is the slope and \(c\) is the intercept.

Step 2: Graph the Equations

Use a graphing utility to graph these functions. Graph \(y = \ln(x - 1) - 2\) and \(y = (9 - 2x) / 3\) on the same set of axes. The point(s) where the graphs intersect are the solutions to the system of equations.

Step 3: Identify the Solutions

Identify the x and y coordinates where the graphs meet. This is the solution to the system of equations. Interpret these results considering the practical context of your problem. If the graphs do not intersect, it implies that there is no solution to the system of equations. If the graphs coincide, it suggests that there are infinite solutions to the system of equations.

Step 4: Confirm the Solution

Verify your answer by substituting the values back into the original equations. Both equations should be satisfied (i.e., the left-hand side should equal the right-hand side) when the solution is applied.

Key Concepts

Using a Graphing Utility

Graphing utilities have revolutionized the way we solve equations. Instead of plotting graphs manually, which can be time-consuming and prone to error, graphing utilities allow students to visualize complex functions quickly and accurately. To use a graphing utility, enter the equations into the system, ensuring that they are in a compatible format. For instance, converting equations into the form of y = mx + b, where m represents the slope and b the y-intercept, makes them ready for graphing.

Once input, the graphing utility will display the graphs on a coordinate plane. Interaction with these utilities is key; you can zoom in or out and adjust the viewing window to find the exact point of intersection between the graphs. It's this intersection that provides the solution to the system of equations. The beauty of graphing utilities is not just in solving equations but also in enabling a visual learning experience that reinforces the understanding of functions and their behaviors.

Understanding the Natural Logarithm

The natural logarithm, denoted as ln, is a mathematical operation that represents the inverse of the exponential function with base e, where e is approximately equal to 2.71828. It's an important concept in various fields of mathematics, including calculus and complex analysis. The natural logarithm of a number x answers the question: To what power must we raise e to obtain x?

When solving systems of equations graphically involving a natural logarithm, like y + 2 = ln(x - 1), it's critical to understand its graph, which is an increasing curve that never touches the x-axis. This property means logarithmic functions will only accept positive arguments, leading to restrictions in the solutions that can be obtained. ln(x - 1) can only be solved for x values greater than 1 since the logarithm of zero or a negative number is undefined.

Solving Linear Equations

Linear equations are the most basic types of algebraic functions and form the foundation for higher-level mathematics. They can be expressed in the general form y = mx + c, with m being the slope of the line and c representing the y-intercept, the point where the line crosses the y-axis. These equations produce straight lines when graphed and have a constant rate of change corresponding to the slope.

Interpreting the Slope and Intercept

Understanding the slope and intercept confers a clear picture of how the graph of a linear equation behaves. A positive slope means the line ascends from left to right, while a negative slope indicates a descent. The y-intercept provides a starting point for the line on the graph. When graphing a system of linear equations to find their point of intersection, you're essentially looking for a common solution that makes both equations true - a pair of x and y values that lie on both lines.