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Chapter 8: Problem 69

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{array}{l} y-e^{-x}=1 \\ y-\ln x=3 \end{array}\right.$$

Short Answer

Expert verified
The solution can be determined graphically due to the transcendental nature of the rearranged equation. The exact solution would depend on the graphing calculator or software used, and should be in the form of an ordered pair (x, y).

Step by step solution

01

Isolate y in both equations

In order to solve the system, it is useful to isolate y in both equations. For the first equation \(y = e^{-x} + 1\) and for the second equation \(y = \ln{x} + 3\). Thus, the system of equations can now be represented as:\[\begin{array}{c} y = e^{-x} + 1 \ y = \ln{x} + 3 \end{array}\]
02

Equate both expressions of y

Since both expressions represent y, they can be equated to each other. This gives us a new equation to solve:\[e^{-x} + 1 = \ln{x} + 3\].
03

Simplifying and rearranging the equation

By moving one of the terms to the other side of the equation, we obtain:\[e^{-x} - \ln{x} = 2.\] This equation is transcendental and usually does not have a straightforward algebraic solution. To solve this equation, we may need numerical methods like Newton's method or graphing methods.
04

Choosing the graphing method

Since our equation is transcendental and doesn't allow for an easy algebraic solution, we opt for the graphical method. The graphical method involves plotting each side of the equation separately and finding the x-value(s) at which they intersect. That is, plot \(y = e^{-x}\) and \(y = \ln{x} + 2\) on the same graph and determine the x-value where these two graphs intersect.
05

Identify the point(s) of intersection

Using graphing software or graphing calculator, plot the graphs and find the point(s) of intersection, which will be the solution for the system. To find the y-value, substitute the x-value(s) obtained into one of the original equations.
06

Verify your solution

Substitute the obtained solution back into the original system of equations to make sure it satisfies both equations. If it doesn't, then recheck your calculation or plotting.

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Most popular questions from this chapter

Write an equation of the line passing through the two points. Use the slope- intercept form, if possible. If not possible, explain why. $$(4,-2),(4,5)$$

Determine whether the statement is true or false. Justify your answer. When the product of two square matrices is the identity matrix, the matrices are inverses of one another.

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Solve for \(x\) $$\left|\begin{array}{cc} 2 x & 1 \\ -1 & x-1 \end{array}\right|=x$$
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