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Chapter 5: Problem 76

Use a graphing utility to approximate the solution(s) to the system of equations. Round the coordinates to 3 decimal places. $$ \begin{array}{l} y=-0.7 x+4 \\ y=\ln x \end{array} $$

Short Answer

Expert verified
The solution is approximately at \( (5.004, 1.610) \).

Step by step solution

01

- Set Up Both Equations

Input the first equation into the graphing utility: \( y = -0.7x + 4 \). Next, input the second equation: \( y = \ln{x} \).
02

- Graph Both Equations

Graph both equations on the same set of axes using the graphing utility. Ensure both lines are clearly visible.
03

- Identify the Intersection Point(s)

Use the graphing utility to find the point(s) where the two graphs intersect. This point is the solution to the system of equations. The coordinates should be roughly estimated to three decimal places.
04

- Record the Solution

Record the coordinates of the intersection point. These coordinates represent the solution to the system of equations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

System of Equations
A system of equations consists of two or more equations that share common variables. Solving them means finding the values of these variables that satisfy all equations simultaneously.
In this exercise, we deal with two equations:
  • Linear: \( y = -0.7x + 4 \)
  • Non-linear (logarithmic): \( y = \ln{x} \)
To solve this system, we look for values of \(x\) and \(y\) that make both equations true at the same time.
Intersection Point
An intersection point is where the graphs of the equations in our system meet. This point's coordinates (x, y) satisfy both equations.
  • First, we graph both equations on the same axes using a graphing utility.
  • Then, we look for the point where they cross each other. This is our intersection point.
For this exercise, we are looking for the intersection of \( y = -0.7x + 4 \) and \( y = \ln{x} \). The coordinates of this point are the solution to the system of equations.
Decimal Approximation
When solving real-world problems or using graphing utilities, we often get solutions that are not exact. Decimal approximation helps in such cases.
Here, we need to approximate the coordinates of the intersection point to three decimal places.
This means if the intersection point is at (1.23456, 2.34567), we round it to (1.235, 2.346). This makes our answer precise enough for practical use while remaining easy to understand.
Graphing Techniques
Using a graphing utility efficiently can save a lot of time and reduce errors. Here鈥檚 how to make the most out of it:
  • First, input your equations clearly and correctly.
  • Check whether your graphing window (X and Y axis limits) shows enough of your graph to see intersections.
  • Use zoom and trace functionalities to closely examine where the graphs intersect.
For example, input \( y = -0.7x + 4 \) and \( y = \ln{x} \) into your graphing utility and ensure both are visible within your chosen window. Then, use the tool to find the intersection coordinates accurately, rounding the values to three decimal places as required.

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Most popular questions from this chapter

A couple has \(\$ 60,000\) to invest for retirement. They plan to put \(x\) dollars in stocks and \(y\) dollars in bonds. For parts (a)-(d), write an inequality to represent the given statement. a. The total amount invested is at most \(\$ 60,000\). b. The couple considers stocks a riskier investment, so they want to invest at least twice as much in bonds as in stocks. c. The amount invested in stocks cannot be negative. d. The amount invested in bonds cannot be negative. e. Graph the solution set to the system of inequalities from parts (a)-(d).

Michelle borrows a total of \(\$ 5000\) in student loans from two lenders. One charges \(4.6 \%\) simple interest and the other charges \(6.2 \%\) simple interest. She is not required to pay off the principal or interest for 3 yr. However, at the end of 3 yr, she will owe a total of \(\$ 762\) for the interest from both loans. How much did she borrow from each lender?

Describe the solution set to the system of inequalities. \(x \geq 0, y \geq 0, x \leq 1, y \leq 1\)

Let \(x\) represent the number of country songs that Sierra puts on a playlist on her portable media player. Let \(y\) represent the number of rock songs that she puts on the playlist. For parts (a)-(e), write an inequality to represent the given statement. a. Sierra will put at least 6 country songs on the playlist. b. Sierra will put no more than 10 rock songs on the playlist. c. Sierra wants to limit the length of the playlist to at most 20 songs. d. The number of country songs cannot be negative. e. The number of rock songs cannot be negative. f. Graph the solution set to the system of inequalities from parts (a)-(e).

Determine if the ordered pair is a solution to the system of equations. (See Example 1\()\) \(y=\frac{3}{2} x-5\) \(6 x-4 y=20\) a. (2,-2) b. (-4,-11)
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