Systems of Linear Equations: Two Variables

For the following exercises, determine whether the given ordered pair is a solution to the system of equations.

1. [latex]-3x - 5y = 13[/latex] and [latex]-x + 4y = 10[/latex] and [latex](-6, 1)[/latex]
2. [latex]-2x + 5y = 7[/latex] and [latex]2x + 9y = 7[/latex] and [latex](-1, 1)[/latex]

For the following exercises, solve each system by substitution.

3. [latex]x + 3y = 5[/latex]
   [latex]2x + 3y = 4[/latex]
4. [latex]4x + 2y = -10[/latex]
   [latex]3x + 9y = 0[/latex]
5. [latex]-2x+3y=1.2[/latex]
   [latex]-3x-6y=1.8[/latex]
6. [latex]3x + 5y = 9[/latex]
   [latex]30x + 50y = -90[/latex]
7. [latex]\frac{1}{2}x + \frac{1}{3}y = 16[/latex]
   [latex]\frac{1}{6}x + \frac{1}{4}y = 9[/latex]

For the following exercises, solve each system by addition.

8. [latex]-2x + 5y = -42[/latex]
   [latex]7x + 2y = 30[/latex]
9. [latex]5x - y = -2.6[/latex]
   [latex]-4x - 6y = 1.4[/latex]
10. [latex]-x + 2y = -1[/latex]
    [latex]5x - 10y = 6[/latex]
11. [latex]\frac{5}{6}x + \frac{1}{4}y = 0[/latex]
    [latex]\frac{1}{8}x - \frac{1}{2}y = -\frac{43}{120}[/latex]
12. [latex]-0.2x + 0.4y = 0.6[/latex]
    [latex]x - 2y = -3[/latex]

For the following exercises, solve each system by any method.

13. [latex]5x + 9y = 16[/latex]
    [latex]x + 2y = 4[/latex]
14. [latex]5x - 2y = 2.25[/latex]
    [latex]7x - 4y = 3[/latex]
15. [latex]7x - 4y = \frac{7}{6}[/latex]
    [latex]2x + 4y = \frac{1}{3}[/latex]
16. [latex]\frac{7}{3}x - \frac{1}{6}y = 2[/latex]
    [latex]-\frac{21}{6}x + \frac{3}{12}y = -3[/latex]
17. [latex]2.2x + 1.3y = -0.1[/latex]
    [latex]4.2x + 4.2y = 2.1[/latex]

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.

18. [latex]3x - y = 0.6[/latex]
    [latex]x - 2y = 1.3[/latex]
19. [latex]x + 2y = 7[/latex]
    [latex]2x + 6y = 12[/latex]
20. [latex]3x - 2y = 5[/latex]
    [latex]-9x + 6y = -15[/latex]

For the following exercises, solve for the desired quantity.

21. An Ethiopian restaurant has a cost of production [latex]C(x) = 11x + 120[/latex] and a revenue function [latex]R(x) = 5x[/latex]. When does the company start to turn a profit?
22. A musician charges [latex]C(x) = 64x + 20,000[/latex] where [latex]x[/latex] is the total number of attendees at the concert. The venue charges [latex]$80[/latex] per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

For the following exercises, use a system of linear equations with two variables and two equations to solve.

23. Find two numbers whose sum is [latex]28[/latex] and difference is [latex]13[/latex].
24. The startup cost for a restaurant is [latex]$120,000[/latex], and each meal costs [latex]$10[/latex] dollars for the restaurant to make. If each meal is then sold for [latex]$15[/latex] dollars, after how many meals does the restaurant break even?
25. A total of [latex]1,595[/latex] first- and second-year college students gathered at a pep rally. The number of first-years exceeded the number of second-years by [latex]15[/latex]. How many students from each year group were in attendance?
26. There were [latex]130[/latex] faculty at a conference. If there were [latex]18[/latex] more women than men attending, how many of each gender attended the conference?
27. If a scientist mixed [latex]10 \%[/latex] saline solution with [latex]60 \%[/latex] saline solution to get [latex]25[/latex] gallons of [latex]40 \%[/latex] saline solution, how many gallons of [latex]10 \%[/latex] and [latex]60 \%[/latex] solutions were mixed?

Systems of Linear Equations: Three Variables

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.

1. [latex]3x + 5y + 2z = 0[/latex] and [latex](3, -3, -5)[/latex]
   [latex]x + y = 0[/latex]
   [latex]6x - 7y + z = 2[/latex]
2. [latex]x - z = 5[/latex] and [latex](4, 4, -1)[/latex]
   [latex]x + y + z = -1[/latex]

For the following exercises, solve each system by elimination.

3. [latex]2x + 4y + z = 16[/latex]
   [latex]2x + 3y + 5z = 20[/latex]
   [latex]5x - 2y + 3z = 20[/latex]
4. [latex]-3x + 2y + z = 10[/latex]
   [latex]4x - 3y + 5z = -3[/latex]
   [latex]4x - 3y + 5z = 31[/latex]
5. [latex]-4x + 6y - 7z = -1[/latex]
   [latex]3x + 2y - z = 4[/latex]

For the following exercises, solve each system by Gaussian elimination.

6. [latex]-5x + 4y - 2z = -46[/latex]
   [latex]2y + 5z = -7[/latex]
   [latex]5x - 6y + 3z = 50[/latex]
7. [latex]-4x - 6y + 12z = -2[/latex]
   [latex]x + 2y + 5z = 10[/latex]
   [latex]4x + 6y - 2z = 8[/latex]
8. [latex]-3x + 2y + z = 11[/latex]
   [latex]-x + 5y + 3z = 4[/latex]
   [latex]-x + 5y + 3z = -4[/latex]
9. [latex]2y + 3z = -14[/latex]
   [latex]-16y-24z = -112[/latex]
   [latex]5x-3y + 4z = -1[/latex]
10. [latex]2x - y + 3z = 0[/latex]
    [latex]x - z = 0[/latex]
    [latex]3x + 2y-5z = 6[/latex]
11. [latex]2x - y + 3z = 0[/latex]
    [latex]x - z = 1[/latex]

Real-World Applications

12. Three even numbers sum up to [latex]108[/latex]. The smaller is half the larger and the middle number is [latex]\dfrac{3}{4}[/latex] the larger. What are the three numbers?

13.  At a family reunion, there were only blood relatives, consisting of children, parents, and grandparents, in attendance. There were [latex]400[/latex] people total. There were twice as many parents as grandparents, and [latex]50[/latex] more children than parents. How many children, parents, and grandparents were in attendance?

14. Your roommate, Shani, offered to buy groceries for you and your other roommate. The total bill was [latex]$82[/latex]. She forgot to save the individual receipts but remembered that your groceries were [latex]$0.05[/latex] cheaper than half of her groceries, and that your other roommate’s groceries were [latex]$2.10[/latex] more than your groceries. How much was each of your share of the groceries?

15. 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 [latex]$82,000[/latex]. The office manager makes [latex]$4,000[/latex] more than the truck driver annually. The annual salaries of the warehouse manager and the truck driver total [latex]$78,000[/latex]. What is the annual salary of each of the co-workers?
16. A local band sells out for their concert. They sell all [latex]1,175[/latex] tickets for a total purse of [latex]$28,112.50[/latex]. The tickets were priced at [latex]$20[/latex] for student tickets, [latex]$22.50[/latex] for children, and [latex]$29[/latex] for adult tickets. If the band sold twice as many adult as children tickets, how many of each type was sold?
17. Last year, at Haven’s Pond Car Dealership, for a particular model of BMW, Jeep, and Toyota, one could purchase all three cars for a total of [latex]$140,000[/latex]. This year, due to inflation, the same cars would cost [latex]$151,830[/latex]. The cost of the BMW increased by [latex]8 \%[/latex], the Jeep by [latex]5 \%[/latex], and the Toyota by [latex]12 \%[/latex]. If the price of last year’s Jeep was [latex]$7,000[/latex] less than the price of last year’s BMW, what was the price of each of the three cars last year?

Systems of Nonlinear Equations and Inequalities

For the following exercises, solve the system of nonlinear equations using substitution.

1. [latex]y = x-3[/latex]
   [latex]x^2 + y^2 = 9[/latex]
2. [latex]y = -x[/latex]
   [latex]x^2 + y^2 = 9[/latex]

For the following exercises, solve the system of nonlinear equations using elimination.

3. [latex]4x^2-9y^2 = 36[/latex]
   [latex]4x^2 + 9y^2 = 36[/latex]
4. [latex]2x^2 + 4y^2 = 4[/latex]
   [latex]2x^2-4y^2 = 25x-10[/latex]
5. [latex]x^2 + y^2 + \dfrac{1}{16} = 2500[/latex]
   [latex]y = 2x^2[/latex]

For the following exercises, use any method to solve the system of nonlinear equations.

6. [latex]-x^2 + y = 2[/latex]
   [latex]-x + y = 2[/latex]
7. [latex]x^2 + y^2 = 1[/latex]
   [latex]y = -x^2[/latex]
8. [latex]9x^2 + 25y^2 = 225[/latex]
   [latex](x-6)^2 + y^2 = 1[/latex]
9. [latex]2x^3 - x^2 = y[/latex]
   [latex]x^2 + y = 0[/latex]

For the following exercises, use any method to solve the nonlinear system.

10. [latex]x^2 - y^2 = 9[/latex]
    [latex]x = 3[/latex]
11. [latex]x^2 - y^2 = 9[/latex]
    [latex]x - y = 0[/latex]
12. [latex]-x^2 + y = 2[/latex]
    [latex]2y = -x[/latex]
13. [latex]x^2 + y^2 = 1[/latex]
    [latex]y^2 = x^2[/latex]
14. [latex]3x^2 - y^2 = 12[/latex]
    [latex](x-1)^2 + y^2 = 1[/latex]

For the following exercise, graph the inequality.

15. [latex]x^2 + y < 9[/latex]

For the following exercises, graph the system of inequalities. Label all points of intersection.

16. [latex]x^2 + y < 1[/latex] [latex]y > 2x[/latex]
17. [latex]x^2 + y^2 < 25[/latex] [latex]3x^2 - y^2 > 12[/latex]
18. [latex]x^2 + 3y^2 > 16[/latex]
    [latex]3x^2 - y^2 < 1[/latex]

For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions.

19. Two numbers add up to [latex]300[/latex]. One number is twice the square of the other number. What are the numbers?
20. A laptop company has discovered their cost and revenue functions for each day: [latex]C(x) = 3x^2-10x + 200[/latex] and [latex]R(x) = -2x^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.

Partial Fraction Decomposition

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.

1. [latex]\dfrac{3x-79}{x^2-5x-24}[/latex]
2. [latex]\dfrac{10x+47}{x^2+7x+10}[/latex]
3. [latex]\dfrac{32x-11}{20x^2-13x+2}[/latex]
4. [latex]\dfrac{5x}{x^2-9}[/latex]

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.

5. [latex]\dfrac{x}{(x-2)^2}[/latex]
6. [latex]\dfrac{-24x-27}{(4x+5)^2}[/latex]
7. [latex]\dfrac{5-x}{(x-7)^2}[/latex]
8. [latex]\dfrac{5x^2+20x+8}{2x(x+1)^2}[/latex]

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

9. [latex]\dfrac{4x^2+6x+11}{(x+2)(x^2+x+3)}[/latex]
10. [latex]\dfrac{-2x^2+10x+4}{(x-1)(x^2+3x+8)}[/latex]
11. [latex]\dfrac{4x^2+17x-1}{(x+3)(x^2+6x+1)}[/latex]
12. [latex]\dfrac{4x^2+5x+3}{x^3-1}[/latex]

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor.

13. [latex]\dfrac{x^3+6x^2+5x+9}{(x^2+1)^2}[/latex]
14. [latex]\dfrac{x^2+5x+5}{(x+2)^2}[/latex]
15. [latex]\dfrac{x^2+25}{(x^2+3x+25)^2}[/latex]
16. [latex]\dfrac{5x+2}{x(x^2+4)^2}[/latex]