MATH 106 6383 Finite Mathematics (2192)

Week 2 Discussion

LINEAR EQUATIONS AND INEQUALITIES (Basic Mathematics Review, Chapter 5, Sections 5.5 – 5.8, and Chapter 7, Sections 7.2 – 7.7)

Solve for the variable:

(c/2) – 8 = 0

5(m – 3) + 4 = -1

3r + 25 = – 1

3x/4 + 2 = 14

-7(2a-1) = 63

3x + 7 = 5x – 21

4x – 36 + 2 = -6

3y/4 + 16 = 10

4(x+2) = 20

-2(a-3) = 16

6x – 17 = 3

x/7 – 15 = -11

-(8r+1) = 33

4y + 5 = -3

2(3x+1) -5x = 4(x-6) + 17

Solve P =R – C for R. Find the value of R when P =480 and C =210.

Solve y =5x +8 for x.

Solve 3y −6x =12 for y.

Solve 4y +2x +8=0 for y.

Solve:

Twenty percent of a number is 6 What is the number?

This year an item costs $106, an increase of 10% over last year’s price. What was last year’s price?

The perimeter of a square is 44 inches. Find the length of a side.

Nine percent of a number is 77.4. What is the number?

Two consecutive integers sum to 63. What are they?

If twenty-one is subtracted from some number and that result is multiplied by two, the result is thirty-eight. What is the number?

If 37% more of a quantity is 159.1, what is the quantity?

A company has determined that it must increase production of a certain line of goods by 112 times last year’s production. The new output will be 2885 items. What was last year’s output?

A lumber company has contracted to cut boards into two pieces so that one piece is three times the length of the other piece. If a board is 12 feet long, what is the length of each piece after cutting?

A television commercial advertises that a certain type of battery will last, on the average, 20 hours longer than twice the life of another type of battery. If consumer tests show that the advertised battery lasts 725 hours, how many hours must the other type of battery last for the advertiser’s claim to be valid?

A statistician is collecting data to help her estimate the number of pickpockets in a certain city. She needs 150 pieces of data and is 34 percent done. How many pieces of data has she collected?

A television commercial advertises that a certain type of light bulb will last, on the average, 200 hours longer than three times the life of another type of bulb. If consumer tests show that the advertised bulb lasts 4700 hours, how many hours must the other type of bulb last for the advertiser’s claim to be valid?

Solve the inequalities for the following exercises. Express solution set using interval notation, and graph the solution set on a number line (see Content > Course Resources > Webliography to see how to create a number line in a word-processing application)

4x+3>23

(– b/3) ≤4

6a– 6≤−27

−8a≤−88

– 11y+4>15

(16c/3)≥−48

−7(8x+10)+2<−32 5x+4≥7x+16 x– 5<3x–11 −(5x+6)+2x– 1<3(1−4x)+11 WRITING AND GRAPHING LINEAR EQUATIONS IN TWO VARIABLES For the following exercises, write the equations of the lines using the given information. Write the equations in slope-intercept form. Slope=−11,y-intercept=−4 m =3,(4,1) Slope=−5,y-intercept=1 m =2,(1,5) m=6,(5,−2) (2,3) , (3,5) m=−5,(2,−3) (4,4),(5,1) m=−9,(−4,−7) (6,1),(5,3) m=−2,(0,2) (8,6) , (7,2) m=−1,(2,0) (−3,1),(2,3) (−1,4),(−2,−4) (0,−5),(6,−1) (2,1),(6,1) (−5,7),(−2,7) (4,1),(4,3) (−1,−1),(−1,5) (0,4),(0,−3) (0,2),(1,0) For the following problems, reading only from the graph, determine the equation of the line. MODELING WITH LINEAR FUNCTIONS (Precalculus, Section 2.4) A town’s population has been decreasing at a constant rate. In 2010 the population was 5,900. By 2012 the population had dropped 4,700. Assume this trend continues. Predict the population in 2016. Identify the year in which the population will reach 0 A town’s population has been increased at a constant rate. In 2010 the population was 46,020. By 2012 the population had increased to 52,070. Assume this trend continues. Predict the population in 2016. Identify the year in which the population will reach 75,000 The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were inflicted. Find the linear function that models the number of people inflicted with the common cold C as a function of the year,t. When will the output reach 0? In what year will the number of people be 9,700? In 2003, a town’s population was 1431. By 2007 the population had grown to 2134. Assume the population is changing linearly. What is the average population growth per year? Find an equation for the population, P of the town t years after 2003. Using your equation, predict the population of the town in 2023. A phone company has a monthly cellular data plan where a customer pays a flat monthly fee of $10 and then a certain amount of money per megabyte (MB) of data used on the phone. If a customer uses 20 MB, the monthly cost will be $11.20. If the customer uses 130 MB, the monthly cost will be $17.80. Find a linear equation for the monthly cost of the data plan as a function of x, the number of MB used. Determine the slope and y-intercept of the equation. Use your equation to find the total monthly cost if 250 MB are used. In 2003, the owl population in a park was measured to be 340. By 2007, the population was measured again to be 285. The population changes linearly. Let the input be years since 2003. Find a formula for the owl population, P. Let the input be yearstsince 2003. What does your model predict the owl population to be in 2019? When hired at a new job selling electronics, you are given two pay options: Option A: Base salary of $14,000 a year with a commission of 10% of your sales Option B: Base salary of $19,000 a year with a commission of 4% of your sales. How much electronics would you need to sell for option A to produce a larger income? FITTING LINEAR MODELS TO DATA / REGRESSION (Precalculus, Section 2.5) The U.S. Census tracks the percentage of persons 25 years or older who are college graduates. That data for several years is given below. Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will the percentage exceed 35%? Year 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 Percent Graduates 21.3 21.4 22.2 23.6 24.4 25.6 26.7 27.7 28 29.4 SOLVING SYSTEMS OF LINEAR EQUATIONS (Precalculus, Section 9.2) For the following exercises, solve each system by substitution. 3x−2y=185x+10y=−10 2x+4y=−3.89x−5y=1.3 x −0.2y=1−10x+2y=5 3x+5y=930x+50y=−90 −3x+y=212x−4y=−8 −x/4+3y/2=11−x/8+y/3=3 For the following exercises, solve each system by addition. −2x+5y=−42 7x+2y=30 6x−5y=−342x+6y=4 5x– y=−2.6−4x−6y=1.4 7x−2y=34x+5y=3.25 –x+2y=−15x−10y=6 7x+6y=2−28x−24y=−8 For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions. 3x– y=0.6 ,x−2y=1.3 –x+2y=4 , 2x−4y=1 x +2y=7 , 2x+6y=12 3x−5y=7 , x−2y=3 3x−2y=5 , −9x+6y=−15 APPLICATIONS OF SYSTEMS OF LINEAR EQUATIONS A guitar factory has a cost of production C(x)=75x+50,000. If the company needs to break even after 150 units sold, at what price should they sell each guitar? Round up to the nearest dollar, Write the revenue functionR(x). A moving company charges a flat rate of $150, and an additional $5 for each box. If a taxi service would charge $20 for each box, how many boxes would you need for it to be cheaper to use the moving company? What would be the total cost?