{"id":2170,"date":"2025-08-07T17:35:33","date_gmt":"2025-08-07T17:35:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2170"},"modified":"2026-04-01T08:09:16","modified_gmt":"2026-04-01T08:09:16","slug":"systems-of-equations-and-inequalities-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-equations-and-inequalities-get-stronger\/","title":{"raw":"Systems of Equations and Inequalities: Get Stronger","rendered":"Systems of Equations and Inequalities: Get Stronger"},"content":{"raw":"

Systems of Linear Equations: Two Variables<\/h2>\r\n1.\u00a0Can a system of linear equations have exactly two solutions? Explain why or why not.\r\n\r\n5. Given a system of equations, explain at least two different methods of solving that system.\r\n\r\nFor the following exercises, determine whether the given ordered pair is a solution to the system of equations.\r\n\r\n7. [latex]\\begin{gathered}-3x - 5y=13 \\\\ -x+4y=10 \\end{gathered}[\/latex] and [latex]\\left(-6,1\\right)[\/latex]\r\n\r\n9. [latex]\\begin{gathered}-2x+5y=7 \\\\ 2x+9y=7 \\end{gathered}[\/latex] and [latex]\\left(-1,1\\right)[\/latex]\r\n\r\nFor the following exercises, solve each system by substitution.\r\n\r\n11. [latex]\\begin{gathered}x+3y=5 \\\\ 2x+3y=4 \\end{gathered}[\/latex]\r\n\r\n13. [latex]\\begin{gathered}4x+2y=-10\\\\ 3x+9y=0\\end{gathered}[\/latex]\r\n\r\n15. [latex]\\begin{gathered}-2x+3y=1.2 \\\\ -3x - 6y=1.8 \\end{gathered}[\/latex]\r\n\r\n17. [latex]\\begin{gathered}3x+5y=9 \\\\ 30x+50y=-90 \\end{gathered}[\/latex]\r\n\r\n19. [latex]\\begin{gathered}\\frac{1}{2}x+\\frac{1}{3}y=16\\\\ \\frac{1}{6}x+\\frac{1}{4}y=9\\end{gathered}[\/latex]\r\n\r\nFor the following exercises, solve each system by addition.\r\n\r\n21. [latex]\\begin{gathered}-2x+5y=-42 \\\\ 7x+2y=30 \\end{gathered}[\/latex]\r\n\r\n23. [latex]\\begin{gathered}5x-y=-2.6 \\\\ -4x - 6y=1.4 \\end{gathered}[\/latex]\r\n\r\n25. [latex]\\begin{gathered}\\mathrm{-x}+2y=-1 \\\\ 5x - 10y=6 \\end{gathered}[\/latex]\r\n\r\n27. [latex]\\begin{gathered}\\frac{5}{6}x+\\frac{1}{4}y=0\\\\ \\frac{1}{8}x-\\frac{1}{2}y=-\\frac{43}{120}\\end{gathered}[\/latex]\r\n\r\n29. [latex]\\begin{gathered}-0.2x+0.4y=0.6 \\\\ x - 2y=-3\\end{gathered}[\/latex]\r\n\r\nFor the following exercises, solve each system by any method.\r\n\r\n31. [latex]\\begin{gathered}5x+9y=16 \\\\ x+2y=4\\end{gathered}[\/latex]\r\n\r\n33. [latex]\\begin{gathered}5x - 2y=2.25\\\\ 7x - 4y=3\\end{gathered}[\/latex]\r\n\r\n35. [latex]\\begin{gathered}7x - 4y=\\frac{7}{6} \\\\ 2x+4y=\\frac{1}{3}\\end{gathered}[\/latex]\r\n\r\n37. [latex]\\begin{gathered}\\frac{7}{3}x-\\frac{1}{6}y=2 \\\\ -\\frac{21}{6}x+\\frac{3}{12}y=-3 \\end{gathered}[\/latex]\r\n\r\n39. [latex]\\begin{gathered}2.2x+1.3y=-0.1\\\\ 4.2x+4.2y=2.1\\end{gathered}[\/latex]\r\n\r\nFor 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.\r\n\r\n41. [latex]\\begin{gathered}3x-y=0.6\\\\ x - 2y=1.3\\end{gathered}[\/latex]\r\n\r\n43. [latex]\\begin{gathered}x+2y=7 \\\\ 2x+6y=12 \\end{gathered}[\/latex]\r\n\r\n45. [latex]\\begin{gathered}3x - 2y=5 \\\\ -9x+6y=-15 \\end{gathered}[\/latex]\r\n\r\nFor the following exercises, use a graphing device to solve each system. Round all answers to the nearest hundredth.\r\n\r\n47. [latex]\\begin{gathered}-0.01x+0.12y=0.62 \\\\ 0.15x+0.20y=0.52\\end{gathered}[\/latex]\r\n\r\n49. [latex]\\begin{gathered}0.15x+0.27y=0.39 \\\\ -0.34x+0.56y=1.8 \\end{gathered}[\/latex]\r\n\r\nFor the following exercises, use a system of linear equations with two variables and two equations to solve.\r\n\r\n61. Find two numbers whose sum is 28 and difference is 13.\r\n\r\n63. The startup cost for a restaurant is $120,000, and each meal costs $10 for the restaurant to make. If each meal is then sold for $15, after how many meals does the restaurant break even?\r\n\r\n65. A total of 1,595 first- and second-year college students gathered at a pep rally. The number of freshmen exceeded the number of sophomores by 15. How many freshmen and sophomores were in attendance?\r\n\r\n67. There were 130 faculty at a conference. If there were 18 more women than men attending, how many of each gender attended the conference?\r\n\r\n69. If a scientist mixed 10% saline solution with 60% saline solution to get 25 gallons of 40% saline solution, how many gallons of 10% and 60% solutions were mixed?\r\n\r\n73. If an investor invests $23,000 into two bonds, one that pays 4% in simple interest, and the other paying 2% simple interest, and the investor earns $710.00 annual interest, how much was invested in each account?\r\n

Systems of Linear Equations: Three Variables<\/h2>\r\n1. Can a linear system of three equations have exactly two solutions? Explain why or why not\r\n\r\nFor the following exercises, determine whether the ordered triple given is the solution to the system of equations.\r\n\r\n7. [latex]\\begin{align}6x-y+3z&=6 \\\\ 3x+5y+2z&=0 \\\\ x+y&=0 \\end{align}[\/latex] and [latex]\\left(3,-3,-5\\right)[\/latex]\r\n\r\n9. [latex]\\begin{align}x-y&=0 \\\\ x-z&=5 \\\\ x-y+z&=-1 \\end{align}[\/latex] and [latex]\\left(4,4,-1\\right)[\/latex]\r\n\r\nFor the following exercises, solve each system by substitution.\r\n\r\n11. [latex]\\begin{align}3x - 4y+2z&=-15 \\\\ 2x+4y+z&=16 \\\\ 2x+3y+5z&=20 \\end{align}[\/latex]\r\n\r\n13. [latex]\\begin{align}5x+2y+4z&=9 \\\\ -3x+2y+z&=10 \\\\ 4x - 3y+5z&=-3 \\end{align}[\/latex]\r\n\r\n15. [latex]\\begin{align}5x - 2y+3z&=4 \\\\ -4x+6y - 7z&=-1 \\\\ 3x+2y-z&=4\\end{align}[\/latex]\r\n\r\nFor the following exercises, solve each system by Gaussian elimination.\r\n\r\n17. [latex]\\begin{align}2x-y+3z&=17 \\\\ -5x+4y - 2z&=-46 \\\\ 2y+5z&=-7 \\end{align}[\/latex]\r\n\r\n19. [latex]\\begin{align}2x+3y - 6z&=1 \\\\ -4x - 6y+12z&=-2 \\\\ x+2y+5z&=10 \\end{align}[\/latex]\r\n\r\n21. [latex]\\begin{align}2x+3y - 4z&=5 \\\\ -3x+2y+z&=11 \\\\ -x+5y+3z&=4 \\end{align}[\/latex]\r\n\r\n23. [latex]\\begin{align}x+y+z&=14 \\\\ 2y+3z&=-14 \\\\ -16y - 24z&=-112 \\end{align}[\/latex]\r\n\r\n25. [latex]\\begin{align}x+y+z&=0 \\\\ 2x-y+3z&=0 \\\\ x-z&=0 \\end{align}[\/latex]\r\n\r\n27. [latex]\\begin{align}x+y+z&=0\\\\ 2x-y+3z&=0 \\\\ x-z&=1 \\end{align}[\/latex]\r\n\r\n29. [latex]\\begin{align}6x - 5y+6z&=38 \\\\ \\frac{1}{5}x-\\frac{1}{2}y+\\frac{3}{5}z&=1 \\\\ -4x-\\frac{3}{2}y-z&=-74 \\end{align}[\/latex]\r\n\r\n39. [latex]\\begin{align}1.1x+0.7y - 3.1z&=-1.79\\\\ 2.1x+0.5y - 1.6z&=-0.13\\\\ 0.5x+0.4y - 0.5z&=-0.07\\end{align}[\/latex]\r\n\r\n43. [latex]\\begin{align}0.5x+0.2y - 0.3z&=1\\\\ 0.4x - 0.6y+0.7z&=0.8\\\\ 0.3x - 0.1y - 0.9z&=0.6\\end{align}[\/latex]\r\n\r\n51. Three even numbers sum up to 108. The smaller is half the larger and the middle number is [latex]\\frac{3}{4}[\/latex] the larger. What are the three numbers?\r\n\r\n53. At a family reunion, there were only blood relatives, consisting of children, parents, and grandparents, in attendance. There were 400 people total. There were twice as many parents as grandparents, and 50 more children than parents. How many children, parents, and grandparents were in attendance?\r\n\r\n55. Your roommate, Sarah, offered to buy groceries for you and your other roommate. The total bill was $82. She forgot to save the individual receipts but remembered that your groceries were $0.05 cheaper than half of her groceries, and that your other roommate\u2019s groceries were $2.10 more than your groceries. How much was each of your share of the groceries?\r\n\r\n57. Three coworkers work for the same employer. Their jobs are warehouse manager, office manager, and truck driver. The sum of the annual salaries of the warehouse manager and office manager is $82,000. The office manager makes $4,000 more than the truck driver annually. The annual salaries of the warehouse manager and the truck driver total $78,000. What is the annual salary of each of the co-workers?\r\n\r\n59. A local band sells out for their concert. They sell all 1,175 tickets for a total purse of $28,112.50. The tickets were priced at $20 for student tickets, $22.50 for children, and $29 for adult tickets. If the band sold twice as many adult as children tickets, how many of each type was sold?\r\n\r\n63. You inherit one million dollars. You invest it all in three accounts for one year. The first account pays 3% compounded annually, the second account pays 4% compounded annually, and the third account pays 2% compounded annually. After one year, you earn $34,000 in interest. If you invest four times the money into the account that pays 3% compared to 2%, how much did you invest in each account?\r\n\r\n65. The top three countries in oil consumption in a certain year are as follows: the United States, Japan, and China. In millions of barrels per day, the three top countries consumed 39.8% of the world\u2019s consumed oil. The United States consumed 0.7% more than four times China\u2019s consumption. The United States consumed 5% more than triple Japan\u2019s consumption. What percent of the world oil consumption did the United States, Japan, and China consume?[footnote]\u201cOil reserves, production and consumption in 2001,\u201d accessed April 6, 2014, http:\/\/scaruffi.com\/politics\/oil.html[\/footnote]\r\n\r\n69. At one time, in the United States, 398 species of animals were on the endangered species list. The top groups were mammals, birds, and fish, which comprised 55% of the endangered species. Birds accounted for 0.7% more than fish, and fish accounted for 1.5% more than mammals. What percent of the endangered species came from mammals, birds, and fish?\r\n

Systems of Nonlinear Equations and Inequalities: Two Variables<\/h2>\r\n3. When you graph a system of inequalities, will there always be a feasible region? If so, explain why. If not, give an example of a graph of inequalities that does not have a feasible region. Why does it not have a feasible region?\r\n\r\nFor the following exercises, solve the system of nonlinear equations using substitution.\r\n\r\n7. [latex]\\begin{gathered}y=x - 3 \\\\ {x}^{2}+{y}^{2}=9 \\end{gathered}[\/latex]\r\n\r\n9. [latex]\\begin{gathered}y=-x \\\\ {x}^{2}+{y}^{2}=9 \\end{gathered}[\/latex]\r\n\r\nFor the following exercises, solve the system of nonlinear equations using elimination.\r\n\r\n11. [latex]\\begin{gathered} 4{x}^{2}-9{y}^{2}=36 \\\\ 4{x}^{2}+9{y}^{2}=36 \\end{gathered}[\/latex]\r\n\r\n13. [latex]\\begin{gathered} 2{x}^{2}+4{y}^{2}=4 \\\\ 2{x}^{2}-4{y}^{2}=25x - 10 \\end{gathered}[\/latex]\r\n\r\n15. [latex]\\begin{gathered}{x}^{2}+{y}^{2}+\\frac{1}{16}=2500\\\\ y=2{x}^{2}\\end{gathered}[\/latex]\r\n\r\nFor the following exercises, use any method to solve the system of nonlinear equations.\r\n\r\n17. [latex]\\begin{gathered} -{x}^{2}+y=2 \\\\ -x+y=2 \\end{gathered}[\/latex]\r\n\r\n19. [latex]\\begin{gathered}{x}^{2}+{y}^{2}=1 \\\\ y=-{x}^{2} \\end{gathered}[\/latex]\r\n\r\n21. [latex]\\begin{gathered}9{x}^{2}+25{y}^{2}=225 \\\\ {\\left(x - 6\\right)}^{2}+{y}^{2}=1 \\end{gathered}[\/latex]\r\n\r\n25. [latex]\\begin{gathered}{x}^{2}-{y}^{2}=9 \\\\ x=3 \\end{gathered}[\/latex]\r\n\r\n27. [latex]\\begin{gathered}{x}^{2}-{y}^{2}=9 \\\\ x-y=0 \\end{gathered}[\/latex]\r\n\r\n33. [latex]\\begin{gathered}3{x}^{2}-{y}^{2}=12 \\\\ {\\left(x - 1\\right)}^{2}+{y}^{2}=1 \\end{gathered}[\/latex]\r\n\r\n35. [latex]\\begin{gathered}3{x}^{2}-{y}^{2}=12 \\\\ {x}^{2}+{y}^{2}=16 \\end{gathered}[\/latex]\r\n\r\nFor the following exercises, graph the system of inequalities. Label all points of intersection.\r\n\r\n41. [latex]\\begin{gathered}{x}^{2}+y<1\\\\ y>2x\\end{gathered}[\/latex]\r\n\r\n43. [latex]\\begin{gathered}{x}^{2}+{y}^{2}<25\\\\ 3{x}^{2}-{y}^{2}>12\\end{gathered}[\/latex]\r\n\r\n45. [latex]\\begin{gathered}{x}^{2}+3{y}^{2}>16\\\\ 3{x}^{2}-{y}^{2}<1\\end{gathered}[\/latex]
\r\n\r\nFor the following exercises, find the solutions to the nonlinear equations with two variables.
\r\n\r\n49. [latex]\\begin{gathered}\\frac{6}{{x}^{2}}-\\frac{1}{{y}^{2}}=8\\\\ \\frac{1}{{x}^{2}}-\\frac{6}{{y}^{2}}=\\frac{1}{8}\\end{gathered}[\/latex]
\r\n\r\n51. [latex]\\begin{gathered}{x}^{2}-xy - 2{y}^{2}-6=0 \\\\ {x}^{2}+{y}^{2}=1 \\end{gathered}[\/latex]
\r\n\r\nFor the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions.\r\n\r\n55. Two numbers add up to 300. One number is twice the square of the other number. What are the numbers?\r\n\r\n57. A laptop company has discovered their cost and revenue functions for each day: [latex]C\\left(x\\right)=3{x}^{2}-10x+200[\/latex] and [latex]R\\left(x\\right)=-2{x}^{2}+100x+50[\/latex]. If they want to make a profit, what is the range of laptops per day that they should produce? Round to the nearest number which would generate profit.\r\n

Partial Fractions<\/h2>\r\n4.\u00a0You are unsure if you correctly decomposed the partial fraction correctly. Explain how you could double-check your answer.\r\n\r\nFor the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.\r\n\r\n7. [latex]\\frac{3x - 79}{{x}^{2}-5x - 24}[\/latex]\r\n\r\n11. [latex]\\frac{32x - 11}{20{x}^{2}-13x+2}[\/latex]\r\n\r\n13. [latex]\\frac{5x}{{x}^{2}-9}[\/latex]\r\n\r\n17. [latex]\\frac{4x - 1}{{x}^{2}-x - 6}[\/latex]\r\n\r\nFor the following exercises, find the decomposition of the partial fraction for the repeating linear factors.\r\n\r\n21. [latex]\\frac{x}{{\\left(x - 2\\right)}^{2}}[\/latex]\r\n\r\n23. [latex]\\frac{-24x - 27}{{\\left(4x+5\\right)}^{2}}[\/latex]\r\n\r\n27. [latex]\\frac{5{x}^{2}+20x+8}{2x{\\left(x+1\\right)}^{2}}[\/latex]\r\n\r\n29. [latex]\\frac{54{x}^{3}+127{x}^{2}+80x+16}{2{x}^{2}{\\left(3x+2\\right)}^{2}}[\/latex]\r\n\r\nFor the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.\r\n\r\n31. [latex]\\frac{4{x}^{2}+6x+11}{\\left(x+2\\right)\\left({x}^{2}+x+3\\right)}[\/latex]\r\n\r\n33. [latex]\\frac{-2{x}^{2}+10x+4}{\\left(x - 1\\right)\\left({x}^{2}+3x+8\\right)}[\/latex]\r\n\r\n37. [latex]\\frac{4{x}^{2}+5x+3}{{x}^{3}-1}[\/latex]\r\n\r\n41. [latex]\\frac{4{x}^{2}+4x+12}{8{x}^{3}-27}[\/latex]\r\n\r\n43. [latex]\\frac{-2{x}^{3}-30{x}^{2}+36x+216}{{x}^{4}+216x}[\/latex]\r\n\r\nFor the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor.\r\n\r\n45. [latex]\\frac{{x}^{3}+6{x}^{2}+5x+9}{{\\left({x}^{2}+1\\right)}^{2}}[\/latex]\r\n\r\n47. [latex]\\frac{{x}^{2}+5x+5}{{\\left(x+2\\right)}^{2}}[\/latex]\r\n\r\n49. [latex]\\frac{{x}^{2}+25}{{\\left({x}^{2}+3x+25\\right)}^{2}}[\/latex]\r\n\r\n51. [latex]\\frac{5x+2}{x{\\left({x}^{2}+4\\right)}^{2}}[\/latex]\r\n\r\n53. [latex]\\frac{2x - 9}{{\\left({x}^{2}-x\\right)}^{2}}[\/latex]\r\n\r\nFor the following exercises, perform the operation and then find the partial fraction decomposition.\r\n\r\n57. [latex]\\frac{7}{x+8}+\\frac{5}{x - 2}-\\frac{x - 1}{{x}^{2}-6x - 16}[\/latex]\r\n\r\n59. [latex]\\frac{2x}{{x}^{2}-16}-\\frac{1 - 2x}{{x}^{2}+6x+8}-\\frac{x - 5}{{x}^{2}-4x}[\/latex]","rendered":"

Systems of Linear Equations: Two Variables<\/h2>\n

1.\u00a0Can a system of linear equations have exactly two solutions? Explain why or why not.<\/p>\n

5. Given a system of equations, explain at least two different methods of solving that system.<\/p>\n

For the following exercises, determine whether the given ordered pair is a solution to the system of equations.<\/p>\n

7. [latex]\\begin{gathered}-3x - 5y=13 \\\\ -x+4y=10 \\end{gathered}[\/latex] and [latex]\\left(-6,1\\right)[\/latex]<\/p>\n

9. [latex]\\begin{gathered}-2x+5y=7 \\\\ 2x+9y=7 \\end{gathered}[\/latex] and [latex]\\left(-1,1\\right)[\/latex]<\/p>\n

For the following exercises, solve each system by substitution.<\/p>\n

11. [latex]\\begin{gathered}x+3y=5 \\\\ 2x+3y=4 \\end{gathered}[\/latex]<\/p>\n

13. [latex]\\begin{gathered}4x+2y=-10\\\\ 3x+9y=0\\end{gathered}[\/latex]<\/p>\n

15. [latex]\\begin{gathered}-2x+3y=1.2 \\\\ -3x - 6y=1.8 \\end{gathered}[\/latex]<\/p>\n

17. [latex]\\begin{gathered}3x+5y=9 \\\\ 30x+50y=-90 \\end{gathered}[\/latex]<\/p>\n

19. [latex]\\begin{gathered}\\frac{1}{2}x+\\frac{1}{3}y=16\\\\ \\frac{1}{6}x+\\frac{1}{4}y=9\\end{gathered}[\/latex]<\/p>\n

For the following exercises, solve each system by addition.<\/p>\n

21. [latex]\\begin{gathered}-2x+5y=-42 \\\\ 7x+2y=30 \\end{gathered}[\/latex]<\/p>\n

23. [latex]\\begin{gathered}5x-y=-2.6 \\\\ -4x - 6y=1.4 \\end{gathered}[\/latex]<\/p>\n

25. [latex]\\begin{gathered}\\mathrm{-x}+2y=-1 \\\\ 5x - 10y=6 \\end{gathered}[\/latex]<\/p>\n

27. [latex]\\begin{gathered}\\frac{5}{6}x+\\frac{1}{4}y=0\\\\ \\frac{1}{8}x-\\frac{1}{2}y=-\\frac{43}{120}\\end{gathered}[\/latex]<\/p>\n

29. [latex]\\begin{gathered}-0.2x+0.4y=0.6 \\\\ x - 2y=-3\\end{gathered}[\/latex]<\/p>\n

For the following exercises, solve each system by any method.<\/p>\n

31. [latex]\\begin{gathered}5x+9y=16 \\\\ x+2y=4\\end{gathered}[\/latex]<\/p>\n

33. [latex]\\begin{gathered}5x - 2y=2.25\\\\ 7x - 4y=3\\end{gathered}[\/latex]<\/p>\n

35. [latex]\\begin{gathered}7x - 4y=\\frac{7}{6} \\\\ 2x+4y=\\frac{1}{3}\\end{gathered}[\/latex]<\/p>\n

37. [latex]\\begin{gathered}\\frac{7}{3}x-\\frac{1}{6}y=2 \\\\ -\\frac{21}{6}x+\\frac{3}{12}y=-3 \\end{gathered}[\/latex]<\/p>\n

39. [latex]\\begin{gathered}2.2x+1.3y=-0.1\\\\ 4.2x+4.2y=2.1\\end{gathered}[\/latex]<\/p>\n

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.<\/p>\n

41. [latex]\\begin{gathered}3x-y=0.6\\\\ x - 2y=1.3\\end{gathered}[\/latex]<\/p>\n

43. [latex]\\begin{gathered}x+2y=7 \\\\ 2x+6y=12 \\end{gathered}[\/latex]<\/p>\n

45. [latex]\\begin{gathered}3x - 2y=5 \\\\ -9x+6y=-15 \\end{gathered}[\/latex]<\/p>\n

For the following exercises, use a graphing device to solve each system. Round all answers to the nearest hundredth.<\/p>\n

47. [latex]\\begin{gathered}-0.01x+0.12y=0.62 \\\\ 0.15x+0.20y=0.52\\end{gathered}[\/latex]<\/p>\n

49. [latex]\\begin{gathered}0.15x+0.27y=0.39 \\\\ -0.34x+0.56y=1.8 \\end{gathered}[\/latex]<\/p>\n

For the following exercises, use a system of linear equations with two variables and two equations to solve.<\/p>\n

61. Find two numbers whose sum is 28 and difference is 13.<\/p>\n

63. The startup cost for a restaurant is $120,000, and each meal costs $10 for the restaurant to make. If each meal is then sold for $15, after how many meals does the restaurant break even?<\/p>\n

65. A total of 1,595 first- and second-year college students gathered at a pep rally. The number of freshmen exceeded the number of sophomores by 15. How many freshmen and sophomores were in attendance?<\/p>\n

67. There were 130 faculty at a conference. If there were 18 more women than men attending, how many of each gender attended the conference?<\/p>\n

69. If a scientist mixed 10% saline solution with 60% saline solution to get 25 gallons of 40% saline solution, how many gallons of 10% and 60% solutions were mixed?<\/p>\n

73. If an investor invests $23,000 into two bonds, one that pays 4% in simple interest, and the other paying 2% simple interest, and the investor earns $710.00 annual interest, how much was invested in each account?<\/p>\n

Systems of Linear Equations: Three Variables<\/h2>\n

1. Can a linear system of three equations have exactly two solutions? Explain why or why not<\/p>\n

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.<\/p>\n

7. [latex]\\begin{align}6x-y+3z&=6 \\\\ 3x+5y+2z&=0 \\\\ x+y&=0 \\end{align}[\/latex] and [latex]\\left(3,-3,-5\\right)[\/latex]<\/p>\n

9. [latex]\\begin{align}x-y&=0 \\\\ x-z&=5 \\\\ x-y+z&=-1 \\end{align}[\/latex] and [latex]\\left(4,4,-1\\right)[\/latex]<\/p>\n

For the following exercises, solve each system by substitution.<\/p>\n

11. [latex]\\begin{align}3x - 4y+2z&=-15 \\\\ 2x+4y+z&=16 \\\\ 2x+3y+5z&=20 \\end{align}[\/latex]<\/p>\n

13. [latex]\\begin{align}5x+2y+4z&=9 \\\\ -3x+2y+z&=10 \\\\ 4x - 3y+5z&=-3 \\end{align}[\/latex]<\/p>\n

15. [latex]\\begin{align}5x - 2y+3z&=4 \\\\ -4x+6y - 7z&=-1 \\\\ 3x+2y-z&=4\\end{align}[\/latex]<\/p>\n

For the following exercises, solve each system by Gaussian elimination.<\/p>\n

17. [latex]\\begin{align}2x-y+3z&=17 \\\\ -5x+4y - 2z&=-46 \\\\ 2y+5z&=-7 \\end{align}[\/latex]<\/p>\n

19. [latex]\\begin{align}2x+3y - 6z&=1 \\\\ -4x - 6y+12z&=-2 \\\\ x+2y+5z&=10 \\end{align}[\/latex]<\/p>\n

21. [latex]\\begin{align}2x+3y - 4z&=5 \\\\ -3x+2y+z&=11 \\\\ -x+5y+3z&=4 \\end{align}[\/latex]<\/p>\n

23. [latex]\\begin{align}x+y+z&=14 \\\\ 2y+3z&=-14 \\\\ -16y - 24z&=-112 \\end{align}[\/latex]<\/p>\n

25. [latex]\\begin{align}x+y+z&=0 \\\\ 2x-y+3z&=0 \\\\ x-z&=0 \\end{align}[\/latex]<\/p>\n

27. [latex]\\begin{align}x+y+z&=0\\\\ 2x-y+3z&=0 \\\\ x-z&=1 \\end{align}[\/latex]<\/p>\n

29. [latex]\\begin{align}6x - 5y+6z&=38 \\\\ \\frac{1}{5}x-\\frac{1}{2}y+\\frac{3}{5}z&=1 \\\\ -4x-\\frac{3}{2}y-z&=-74 \\end{align}[\/latex]<\/p>\n

39. [latex]\\begin{align}1.1x+0.7y - 3.1z&=-1.79\\\\ 2.1x+0.5y - 1.6z&=-0.13\\\\ 0.5x+0.4y - 0.5z&=-0.07\\end{align}[\/latex]<\/p>\n

43. [latex]\\begin{align}0.5x+0.2y - 0.3z&=1\\\\ 0.4x - 0.6y+0.7z&=0.8\\\\ 0.3x - 0.1y - 0.9z&=0.6\\end{align}[\/latex]<\/p>\n

51. Three even numbers sum up to 108. The smaller is half the larger and the middle number is [latex]\\frac{3}{4}[\/latex] the larger. What are the three numbers?<\/p>\n

53. At a family reunion, there were only blood relatives, consisting of children, parents, and grandparents, in attendance. There were 400 people total. There were twice as many parents as grandparents, and 50 more children than parents. How many children, parents, and grandparents were in attendance?<\/p>\n

55. Your roommate, Sarah, offered to buy groceries for you and your other roommate. The total bill was $82. She forgot to save the individual receipts but remembered that your groceries were $0.05 cheaper than half of her groceries, and that your other roommate\u2019s groceries were $2.10 more than your groceries. How much was each of your share of the groceries?<\/p>\n

57. Three coworkers work for the same employer. Their jobs are warehouse manager, office manager, and truck driver. The sum of the annual salaries of the warehouse manager and office manager is $82,000. The office manager makes $4,000 more than the truck driver annually. The annual salaries of the warehouse manager and the truck driver total $78,000. What is the annual salary of each of the co-workers?<\/p>\n

59. A local band sells out for their concert. They sell all 1,175 tickets for a total purse of $28,112.50. The tickets were priced at $20 for student tickets, $22.50 for children, and $29 for adult tickets. If the band sold twice as many adult as children tickets, how many of each type was sold?<\/p>\n

63. You inherit one million dollars. You invest it all in three accounts for one year. The first account pays 3% compounded annually, the second account pays 4% compounded annually, and the third account pays 2% compounded annually. After one year, you earn $34,000 in interest. If you invest four times the money into the account that pays 3% compared to 2%, how much did you invest in each account?<\/p>\n

65. The top three countries in oil consumption in a certain year are as follows: the United States, Japan, and China. In millions of barrels per day, the three top countries consumed 39.8% of the world\u2019s consumed oil. The United States consumed 0.7% more than four times China\u2019s consumption. The United States consumed 5% more than triple Japan\u2019s consumption. What percent of the world oil consumption did the United States, Japan, and China consume?[1]<\/sup><\/a><\/p>\n

69. At one time, in the United States, 398 species of animals were on the endangered species list. The top groups were mammals, birds, and fish, which comprised 55% of the endangered species. Birds accounted for 0.7% more than fish, and fish accounted for 1.5% more than mammals. What percent of the endangered species came from mammals, birds, and fish?<\/p>\n

Systems of Nonlinear Equations and Inequalities: Two Variables<\/h2>\n

3. When you graph a system of inequalities, will there always be a feasible region? If so, explain why. If not, give an example of a graph of inequalities that does not have a feasible region. Why does it not have a feasible region?<\/p>\n

For the following exercises, solve the system of nonlinear equations using substitution.<\/p>\n

7. [latex]\\begin{gathered}y=x - 3 \\\\ {x}^{2}+{y}^{2}=9 \\end{gathered}[\/latex]<\/p>\n

9. [latex]\\begin{gathered}y=-x \\\\ {x}^{2}+{y}^{2}=9 \\end{gathered}[\/latex]<\/p>\n

For the following exercises, solve the system of nonlinear equations using elimination.<\/p>\n

11. [latex]\\begin{gathered} 4{x}^{2}-9{y}^{2}=36 \\\\ 4{x}^{2}+9{y}^{2}=36 \\end{gathered}[\/latex]<\/p>\n

13. [latex]\\begin{gathered} 2{x}^{2}+4{y}^{2}=4 \\\\ 2{x}^{2}-4{y}^{2}=25x - 10 \\end{gathered}[\/latex]<\/p>\n

15. [latex]\\begin{gathered}{x}^{2}+{y}^{2}+\\frac{1}{16}=2500\\\\ y=2{x}^{2}\\end{gathered}[\/latex]<\/p>\n

For the following exercises, use any method to solve the system of nonlinear equations.<\/p>\n

17. [latex]\\begin{gathered} -{x}^{2}+y=2 \\\\ -x+y=2 \\end{gathered}[\/latex]<\/p>\n

19. [latex]\\begin{gathered}{x}^{2}+{y}^{2}=1 \\\\ y=-{x}^{2} \\end{gathered}[\/latex]<\/p>\n

21. [latex]\\begin{gathered}9{x}^{2}+25{y}^{2}=225 \\\\ {\\left(x - 6\\right)}^{2}+{y}^{2}=1 \\end{gathered}[\/latex]<\/p>\n

25. [latex]\\begin{gathered}{x}^{2}-{y}^{2}=9 \\\\ x=3 \\end{gathered}[\/latex]<\/p>\n

27. [latex]\\begin{gathered}{x}^{2}-{y}^{2}=9 \\\\ x-y=0 \\end{gathered}[\/latex]<\/p>\n

33. [latex]\\begin{gathered}3{x}^{2}-{y}^{2}=12 \\\\ {\\left(x - 1\\right)}^{2}+{y}^{2}=1 \\end{gathered}[\/latex]<\/p>\n

35. [latex]\\begin{gathered}3{x}^{2}-{y}^{2}=12 \\\\ {x}^{2}+{y}^{2}=16 \\end{gathered}[\/latex]<\/p>\n

For the following exercises, graph the system of inequalities. Label all points of intersection.<\/p>\n

41. [latex]\\begin{gathered}{x}^{2}+y<1\\\\ y>2x\\end{gathered}[\/latex]<\/p>\n

43. [latex]\\begin{gathered}{x}^{2}+{y}^{2}<25\\\\ 3{x}^{2}-{y}^{2}>12\\end{gathered}[\/latex]<\/p>\n

45. [latex]\\begin{gathered}{x}^{2}+3{y}^{2}>16\\\\ 3{x}^{2}-{y}^{2}<1\\end{gathered}[\/latex]<\/p>\n

For the following exercises, find the solutions to the nonlinear equations with two variables.<\/p>\n

49. [latex]\\begin{gathered}\\frac{6}{{x}^{2}}-\\frac{1}{{y}^{2}}=8\\\\ \\frac{1}{{x}^{2}}-\\frac{6}{{y}^{2}}=\\frac{1}{8}\\end{gathered}[\/latex]<\/p>\n

51. [latex]\\begin{gathered}{x}^{2}-xy - 2{y}^{2}-6=0 \\\\ {x}^{2}+{y}^{2}=1 \\end{gathered}[\/latex]<\/p>\n

For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions.<\/p>\n

55. Two numbers add up to 300. One number is twice the square of the other number. What are the numbers?<\/p>\n

57. A laptop company has discovered their cost and revenue functions for each day: [latex]C\\left(x\\right)=3{x}^{2}-10x+200[\/latex] and [latex]R\\left(x\\right)=-2{x}^{2}+100x+50[\/latex]. If they want to make a profit, what is the range of laptops per day that they should produce? Round to the nearest number which would generate profit.<\/p>\n

Partial Fractions<\/h2>\n

4.\u00a0You are unsure if you correctly decomposed the partial fraction correctly. Explain how you could double-check your answer.<\/p>\n

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.<\/p>\n

7. [latex]\\frac{3x - 79}{{x}^{2}-5x - 24}[\/latex]<\/p>\n

11. [latex]\\frac{32x - 11}{20{x}^{2}-13x+2}[\/latex]<\/p>\n

13. [latex]\\frac{5x}{{x}^{2}-9}[\/latex]<\/p>\n

17. [latex]\\frac{4x - 1}{{x}^{2}-x - 6}[\/latex]<\/p>\n

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.<\/p>\n

21. [latex]\\frac{x}{{\\left(x - 2\\right)}^{2}}[\/latex]<\/p>\n

23. [latex]\\frac{-24x - 27}{{\\left(4x+5\\right)}^{2}}[\/latex]<\/p>\n

27. [latex]\\frac{5{x}^{2}+20x+8}{2x{\\left(x+1\\right)}^{2}}[\/latex]<\/p>\n

29. [latex]\\frac{54{x}^{3}+127{x}^{2}+80x+16}{2{x}^{2}{\\left(3x+2\\right)}^{2}}[\/latex]<\/p>\n

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.<\/p>\n

31. [latex]\\frac{4{x}^{2}+6x+11}{\\left(x+2\\right)\\left({x}^{2}+x+3\\right)}[\/latex]<\/p>\n

33. [latex]\\frac{-2{x}^{2}+10x+4}{\\left(x - 1\\right)\\left({x}^{2}+3x+8\\right)}[\/latex]<\/p>\n

37. [latex]\\frac{4{x}^{2}+5x+3}{{x}^{3}-1}[\/latex]<\/p>\n

41. [latex]\\frac{4{x}^{2}+4x+12}{8{x}^{3}-27}[\/latex]<\/p>\n

43. [latex]\\frac{-2{x}^{3}-30{x}^{2}+36x+216}{{x}^{4}+216x}[\/latex]<\/p>\n

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor.<\/p>\n

45. [latex]\\frac{{x}^{3}+6{x}^{2}+5x+9}{{\\left({x}^{2}+1\\right)}^{2}}[\/latex]<\/p>\n

47. [latex]\\frac{{x}^{2}+5x+5}{{\\left(x+2\\right)}^{2}}[\/latex]<\/p>\n

49. [latex]\\frac{{x}^{2}+25}{{\\left({x}^{2}+3x+25\\right)}^{2}}[\/latex]<\/p>\n

51. [latex]\\frac{5x+2}{x{\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/p>\n

53. [latex]\\frac{2x - 9}{{\\left({x}^{2}-x\\right)}^{2}}[\/latex]<\/p>\n

For the following exercises, perform the operation and then find the partial fraction decomposition.<\/p>\n

57. [latex]\\frac{7}{x+8}+\\frac{5}{x - 2}-\\frac{x - 1}{{x}^{2}-6x - 16}[\/latex]<\/p>\n

59. [latex]\\frac{2x}{{x}^{2}-16}-\\frac{1 - 2x}{{x}^{2}+6x+8}-\\frac{x - 5}{{x}^{2}-4x}[\/latex]<\/p>\n

  1. \u201cOil reserves, production and consumption in 2001,\u201d accessed April 6, 2014, http:\/\/scaruffi.com\/politics\/oil.html ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":67,"menu_order":33,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":131,"module-header":"practice","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2170"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":15,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2170\/revisions"}],"predecessor-version":[{"id":6082,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2170\/revisions\/6082"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/131"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2170\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2170"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2170"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2170"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2170"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}