/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 26 The demand equation for the Drak... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 26

The demand equation for the Drake GPS Navigator is \(x+4 p-800=0\), where \(x\) is the quantity demanded per week and \(p\) is the wholesale unit price in dollars. The supply equation is \(x-20 p+\) \(1000=0\), where \(x\) is the quantity the supplier will make available in the market each week when the wholesale price is \(p\) dollars each. Find the equilibrium quantity and the equilibrium price for the GPS Navigators.

Short Answer

Expert verified
The equilibrium quantity (x) for the Drake GPS Navigator is 500 units per week, and the equilibrium price (p) is $75 per unit.

Step by step solution

01

Set up the system of equations

We are given the demand and supply equations of the GPS Navigators: Demand equation: \(x + 4p - 800 = 0\) Supply equation: \(x - 20p + 1000 = 0\) Now we need to solve this system of equations to find the equilibrium quantity (x) and price (p).
02

Solve for x in the demand equation

Rewrite the demand equation to isolate x: Demand equation: \(x = -4p + 800\)
03

Substitute the expression of x from the demand equation into the supply equation

Substitute the expression of x from the demand equation into the supply equation: \(-4p + 800 - 20p + 1000 = 0\)
04

Solve for p

Combine like terms and solve for p: \(-4p - 20p = -1800\) \(-24p = -1800\) Now, divide both sides by -24: \(p = 75\)
05

Find the equilibrium quantity (x)

Substitute the value of p into the expression of x from the demand equation: \(x = -4(75) + 800\) \(x = -300 + 800\) \(x = 500\)
06

Write the final solution

The equilibrium quantity (x) is 500 GPS Navigators, and the equilibrium price (p) is $75 per GPS Navigator.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Demand and Supply Equations
When analyzing markets, economists use demand and supply equations to determine the interaction between consumers buying products (demand) and producers selling them (supply). The demand equation typically represents how much of a product consumers want to purchase at various prices. In contrast, the supply equation shows how much product producers are willing to sell at different price points.

For the Drake GPS Navigator, the equations given are:
  • Demand Equation: ( x + 4p - 800 = 0), where x is the quantity demanded, and p is the price.
  • Supply Equation: (x - 20p + 1000 = 0), which tells us the supplier's available quantity at the price p.
Finding the point where these two equations intersect will give us the equilibrium, where the amount supplied is equal to the amount demanded. This is the point of balance in the market.
Systems of Equations
A system of equations consists of two or more equations with a common set of variables. The goal is to find the values of these variables that satisfy all equations in the system simultaneously. In our example, we have a system with two equations, one representing demand and the other supply, that we need to solve together.Solving a system can be done using various methods, including substitution, elimination, or graphing. In the case of the Drake GPS Navigator problem, substituting the expression for x from the demand equation into the supply equation is the chosen method. This reduces the system to a single equation with one variable (p), which can then be solved algebraically.
Algebraic Problem-Solving
The process of algebraic problem-solving involves manipulating equations to isolate variables and find their values. Starting with rewriting equations in a simpler form to identify relationships between variables is often the first step.

For instance, in our GPS Navigator problem, once we rewrite the demand equation to isolate x, we then substitute this expression into the supply equation, combining like terms to solve for p. Once the value of p is found, we can use it to find x, the equilibrium quantity. The ability to simplify and solve these algebraic expressions is fundamental to understanding the underlying mathematics of economics and many other fields.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The Venus Health Club for Women provides its members with the following table, which gives the average desirable weight (in pounds) for women of a given height (in inches): $$ \begin{array}{lrrrrr} \hline \text { Height, } \boldsymbol{x} & 60 & 63 & 66 & 69 & 72 \\ \hline \text { Weight, } \boldsymbol{y} & 108 & 118 & 129 & 140 & 152 \\ \hline \end{array} $$ a. Plot the weight \((y)\) versus the height \((x)\). b. Draw a straight line \(L\) through the points corresponding to heights of \(5 \mathrm{ft}\) and \(6 \mathrm{ft}\). c. Derive an equation of the line \(L\). d. Using the equation of part (c), estimate the average desirable weight for a woman who is \(5 \mathrm{ft}, 5\) in. tall.

Show that two distinct lines with equations \(a_{1} x+b_{1} y+\) \(c_{1}=0\) and \(a_{2} x+b_{2} y+c_{2}=0\), respectively, are parallel if and only if \(a_{1} b_{2}-b_{1} a_{2}=0\). Hint: Write each equation in the slope-intercept form and compare.

With computer security always a hot-button issue, demand is growing for technology that authenticates and authorizes computer users. The following table gives the authentication software sales (in billions of dollars) from 1999 through \(2004(x=0\) represents 1999): $$ \begin{array}{ccccccc} \hline \text { Year, } \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Sales, } \boldsymbol{y} & 2.4 & 2.9 & 3.7 & 4.5 & 5.2 & 6.1 \\ \hline \end{array} $$ a. Find an equation of the least-squares line for these data. b. Use the result of part (a) to estimate the sales for 2007 , assuming the trend continues.

Metro Department Store's annual sales (in millions of dollars) during the past 5 yr were. $$ \begin{array}{lccccc} \hline \text { Annual Sales, } y & 5.8 & 6.2 & 7.2 & 8.4 & 9.0 \\ \hline \text { Year, } \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 \\ \hline \end{array} $$ a. Plot the annual sales \((y)\) versus the year \((x)\). b. Draw a straight line \(L\) through the points corresponding to the first and fifth years. c. Derive an equation of the line \(L\). d. Using the equation found in part (c), estimate Metro's annual sales 4 yr from now \((x=9)\).

Find an equation of the line that passes through the point \((-2,2)\) and is parallel to the line \(2 x-4 y-8=0\).
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.