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    graphing and various equation problems

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    1.) The national cost "C" in billions of dollars for dental services can be modeled by the linear equation:

    C = 2.85n+ 30.52

    where n is the number of years since 1990 (Health Care Financing Administration, www.hcfa.gov).

    a) Find and interpret the C-intercept for the line.
    Put n = 0, we will get
    C = 30.52

    C-intercept is (0, 30.52)

    b) Find and interpret the n-intercept for the line.
    Put C = 0, we will get
    0 = 2.85n + 30.52
    2.85n = - 30.52
    n = -30.52/2.85 = -10.71(approximately)
    n-intercept is (-10.71, 0)

    c) Graph the line for n ranging from 0 through 20.

    2.) Demand equation. Helen's Health Foods usually sells 400 cans of ProPac Muscle Punch per week when the price is $5 per can. After experimenting with prices for some time, Helen has determined that the weekly demand can be found by using the equation:

    d=600-40p

    where d is the number of cans and p is the price per can.

    a) Will Helen sell more or less Muscle Punch if she raises
    her price from $5?

    If she raises the price from $5, the demand will be less.

    Let us take p = $6, then

    d = 600 - 40*6 = 600 - 240 = 360

    Thus the sales will decrease.

    b) What happens to her sales every time she raises her
    price by $1?

    At every $1 increase in price the sales will decrease by 40.

    c) Graph the equation.

    3.) Marginal revenue. A defense attorney charges her client $4000 plus $120 per hour. The formula R=120n+4000 gives her revenue in dollars for n hours of work.

    a) What is her revenue for 100 hours of work?
    Put n = 100, we will get
    R = 120*100 + 4000 = 12000 + 4000 = 16000

    b) What is her revenue for 101 hours of work?

    Put n = 101, we will get
    R = 120*101 + 4000 = 12120 + 4000 = 16120

    c) By how much did the one extra hour of work increase the revenue? (The increase in revenue is called the marginal revenue for the 101st hour.)

    Increase = 16120 - 16000 = 120

    4.) Gas laws. The volume of a gas is inversely proportional to the pressure on the gas. If the volume is 6 cubic centimeters when the pressure on the gas is 8 kilograms per square centimeter, then what is the volume when the pressure is 12 kilograms per square centimeter?

    Let the volume be V and pressure be d then V = kd

    Given, if the volume is 6 cubic centimeters when the pressure on the gas is 8 kilograms per square centimeter.

    6 = 8k

    => k = 6/8 = ¾

    Now, we have to find the volume when the pressure is 12 kilograms per square centimeter.

    d = 12, k = ¾
    V = kd = (3/4)*12 = 9 cubic centimeters.

    5.) Graph the linear inequality:

    2x<3y+6

    6.) Solve each system by substitution. Determine whether the equations are independent, dependent or inconsistent.

    a) y=-3x+19
    y=2x-1

    Substitute value of y = 2x - 1 in y= -3x+19, we will get

    2x - 1 = -3x + 19
    5x = 20
    X = 4

    Now substitute value of x in y = 2x - 1
    y = 2*4 - 1 = 8 - 1 = 7

    Solution: (4, 7)

    System is consistent and independent as it has unique solution.

    b) y= -4x -7
    y=3x

    Substitute y = 3x in y = -4x - 7, we will get

    3x = -4x - 7

    7x = -7
    x = -1

    y = 3x = -3

    The solution is (-1, -3).
    System is consistent and independent as it has unique solution.

    7) Solve each system by Addition Method

    a) 3x+5y = -11-----------(i)
    x-2y = 11-------------(ii)

    Mutiply equation (ii) by -3

    -3x + 6y = -33

    And add to equation (i)

    3x + 5y = -11

    11y = -44

    y = -4

    Given x - 2y = 11

    x + 8 = 11

    x = 3

    The solution is (3, -4).
    b) 2x=2-4
    3x+y=-1

    EQUATION NOT CLEAR

    8) Solve each system by the Addition Method. Determine whether the equations are independent, dependent or inconsistent.

    a) x-y=3
    -6x+6y=17

    6x - 6y = 18

    -6x + 6y = 17

    No solution.

    Inconsistent and independent

    9) Solve each system by Addition Method

    a)

    Multiply both equations by 63, we will get
    27x + 35y = 1701---------(iii)
    7x + 18y = 441----------(iv)

    Multiply now (iii) by 7 and (iv) by -27
    189x + 245y = 11907
    -189x - 486y = - 11907

    After adding we will get y = 0

    (3/7)x + 0 = 27

    x = 63

    Solution (63, 0)

    b) 3x-2.5y=7.125 --------------(i)
    2.5x-3y=7.3125-----------------(ii)

    Multiply equation (i) by -2.5
    -7.5x + 6.25 y = - 17.8125

    Multiply equation (ii) by 3
    7.5x - 9y=7.3125 = 21.9375
    add
    -2.75y = 4.125

    y = -1.5

    2.5x + 4.5 = 7.3125

    2.5x = 2.8125
    x = 1.125

    Solution (1.125, -1.5)

    © BrainMass Inc. brainmass.com October 9, 2019, 10:32 pm ad1c9bdddf
    https://brainmass.com/math/linear-algebra/graphing-and-various-equation-problems-223168

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    Dear Student,

    I have attached the solution.

    Sanjay
    106417

    1.) The national cost "C" in billions of dollars for dental services can be modeled by the linear equation:

    C = 2.85n+ 30.52

    where n is the number of years since 1990 (Health Care Financing Administration, www.hcfa.gov).

    a) Find and interpret the C-intercept for the line.
    Put n = 0, we will get
    C = 30.52

    C-intercept is (0, 30.52)

    b) Find and interpret the n-intercept for the line.
    Put C = 0, we will get
    0 = 2.85n + 30.52
    2.85n = - 30.52
    n = -30.52/2.85 = -10.71(approximately)
    n-intercept is (-10.71, 0)

    c) Graph the line for n ranging from 0 through 20.

    2.) Demand equation. Helen's Health Foods usually sells 400 cans of ProPac Muscle Punch per week when the price is $5 per can. After experimenting with prices for some time, Helen has determined that the weekly demand can be found by using the equation:

    d=600-40p

    where d is the number of cans and p is the price per can.

    a) Will Helen sell more or less Muscle Punch if she raises
    her price from $5?

    If she raises the price from $5, the demand will be ...

    Solution Summary

    This solution is comprised of a detailed explanation to answer graphing and various equation problems.

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