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Mathematics LibreTexts

4.1E: Exercises

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    30839
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    Practice Makes Perfect

    Determine Whether an Ordered Pair is a Solution of a System of Equations

    In the following exercises, determine if the following points are solutions to the given system of equations.

    1. \(\left\{ \begin{array} {l} 2x−6y=0 \\ 3x−4y=5 \end{array} \right.\)

    ⓐ \((3,1)\)
    ⓑ \((−3,4)\)

    Answer

    ⓐ yes ⓑ no

    2. \(\left\{ \begin{array} {l} −3x+y=8 \\ −x+2y=−9 \end{array} \right.\)

    ⓐ \((−5,−7)\)
    ⓑ \((−5,7)\)

    3. \(\left\{ \begin{array} {l} x+y=2 \\ y=\frac{3}{4}x \end{array} \right.\)

    ⓐ \((87,67)\)
    ⓑ \((1,34)\)

    Answer

    ⓐ yes ⓑ no

    4. \(\left\{ \begin{array} {l} 2x+3y=6 \\ y=\frac{2}{3}x+2 \end{array} \right.\)

    ⓐ \((−6,2)\)
    ⓑ \((−3,4)\)

    Solve a System of Linear Equations by Graphing

    In the following exercises, solve the following systems of equations by graphing.

    5. \(\left\{ \begin{array} {l} 3x+y=−3 \\ 2x+3y=5 \end{array} \right.\)

    Answer

    \((−3,2)\)

    6. \(\left\{ \begin{array} {l} −x+y=2 \\ 2x+y=−4 \end{array} \right.\)

    7. \(\left\{ \begin{array} {l} y=x+2 \\ y=−2x+2 \end{array} \right.\)

    Answer

    \((0,2)\)

    8. \(\left\{ \begin{array} {l} y=x−2 \\ y=−3x+2 \end{array} \right.\)

    9. \(\left\{ \begin{array} {l} y=\frac{3}{2}x+1 \\ y=−\frac{1}{2}x+5 \end{array} \right.\)

    Answer

    \((2,4)\)

    10. \(\left\{ \begin{array} {l} y=\frac{2}{3}x−2 \\ y=−\frac{1}{3}x−5 \end{array} \right.\)

    11. \(\left\{ \begin{array} {l} x+y=−4 \\ −x+2y=−2 \end{array} \right.\)

    Answer

    \((−2,2)\)

    12. \(\left\{ \begin{array} {l} −x+3y=3 \\ x+3y=3 \end{array} \right.\)

    13. \(\left\{ \begin{array} {l} −2x+3y=3 \\ x+3y=12 \end{array} \right.\)

    Answer

    \((3,3)\)

    14. \(\left\{ \begin{array} {l} 2x−y=4 \\ 2x+3y=12 \end{array} \right.\)

    15. \(\left\{ \begin{array} {l} x+3y=−6 \\ y=−\frac{4}{3}x+4 \end{array} \right.\)

    Answer

    \((6,−4)\)

    16. \(\left\{ \begin{array} {l} −x+2y=−6 \\ y=−\frac{1}{2}x−1 \end{array} \right.\)

    17. \(\left\{ \begin{array} {l} −2x+4y=4 \\ y=12x \end{array} \right.\)

    Answer

    no solution

    18. \(\left\{ \begin{array} {l} 3x+5y=10 \\ y=−\frac{3}{5}x+1 \end{array} \right.\)

    19. \(\left\{ \begin{array} {l} 4x−3y=8 \\ 8x−6y=14 \end{array} \right.\)

    Answer

    no solution

    20. \(\left\{ \begin{array} {l} x+3y=4 \\ −2x−6y=3 \end{array} \right.\)

    21. \(\left\{ \begin{array} {l} x=−3y+4 \\ 2x+6y=8 \end{array} \right.\)

    Answer

    infinite solutions with solution set: \(\big\{ (x,y) | 2x+6y=8 \big\}\)

    22. \(\left\{ \begin{array} {l} 4x=3y+7 \\ 8x−6y=14 \end{array} \right.\)

    23. \(\left\{ \begin{array} {l} 2x+y=6 \\ −8x−4y=−24 \end{array} \right.\)

    Answer

    infinite solutions with solution set: \(\big\{ (x,y) | 2x+y=6 \big\}\)

    24. \(\left\{ \begin{array} {l} 5x+2y=7 \\ −10x−4y=−14 \end{array} \right.\)

    Without graphing, determine the number of solutions and then classify the system of equations.

    25. \(\left\{ \begin{array} {l} y=\frac{2}{3}x+1 \\ −2x+3y=5 \end{array} \right.\)

    Answer

    1 point, consistent and independent

    26. \(\left\{ \begin{array} {l} y=\frac{3}{2}x+1 \\ 2x−3y=7 \end{array} \right.\)

    27. \(\left\{ \begin{array} {l} 5x+3y=4 \\ 2x−3y=5 \end{array} \right.\)

    Answer

    1 point, consistent and independent

    28. \(\left\{ \begin{array} {l} y=−12x+5 \\ x+2y=10 \end{array} \right.\)

    29. \(\left\{ \begin{array} {l} 5x−2y=10 \\ y=52x−5 \end{array} \right.\)

    Answer

    infinite solutions, consistent, dependent

    Solve a System of Equations by Substitution

    In the following exercises, solve the systems of equations by substitution.

    30. \(\left\{ \begin{array} {l} 2x+y=−4 \\ 3x−2y=−6\end{array} \right.\)

    31. \(\left\{ \begin{array} {l} 2x+y=−2\\ 3x−y=7 \end{array} \right.\)

    Answer

    \((1,−4)\)

    32. \(\left\{ \begin{array} {l} x−2y=−5 \\ 2x−3y=−4 \end{array} \right.\)

    33. \(\left\{ \begin{array} {l} x−3y=−9 \\ 2x+5y=4 \end{array} \right.\)

    Answer

    \((−3,2)\)

    34. \(\left\{ \begin{array} {l} 5x−2y=−6 \\ y=3x+3 \end{array} \right.\)

    35. \(\left\{ \begin{array} {l} −2x+2y=6 \\ y=−3x+1 \end{array} \right.\)

    Answer

    \((−1/2,5/2)\)

    36. \(\left\{ \begin{array} {l} 2x+5y=1 \\ y=\frac{1}{3}x−2 \end{array} \right.\)

    37. \(\left\{ \begin{array} {l} 3x+4y=1 \\ y=−\frac{2}{5}x+2 \end{array} \right.\)

    Answer

    \((−5,4)\)

    38. \(\left\{ \begin{array} {l} 2x+y=5 \\ x−2y=−15 \end{array} \right.\)

    39. \(\left\{ \begin{array} {l} 4x+y=10 \\ x−2y=−20 \end{array} \right.\)

    Answer

    \((0,10)\)

    40. \(\left\{ \begin{array} {l} y=−2x−1 \\ y=−\frac{1}{3}x+4 \end{array} \right.\)

    41. \(\left\{ \begin{array} {l} y=x−6 \\ y=−\frac{3}{2}x+4 \end{array} \right.\)

    Answer

    \((4,−2)\)

    42. \(\left\{ \begin{array} {l} x=2y \\ 4x−8y=0 \end{array} \right.\)

    43. \(\left\{ \begin{array} {l} 2x−16y=8 \\ −x−8y=−4 \end{array} \right.\)

    Answer

    \((4,0)\)

    44. \(\left\{ \begin{array} {l} y=\frac{7}{8}x+4 \\ −7x+8y=6 \end{array} \right.\)

    45. \(\left\{ \begin{array} {l} y=−\frac{2}{3}x+5 \\ 2x+3y=11 \end{array} \right.\)

    Answer

    no solution

    Solve a System of Equations by Elimination

    In the following exercises, solve the systems of equations by elimination.

    46. \(\left\{ \begin{array} {l} 5x+2y=2 \\ −3x−y=0 \end{array} \right.\)

    47. \(\left\{ \begin{array} {l} 6x−5y=−1 \\ 2x+y=13 \end{array} \right.\)

    Answer

    \((4,5)\)

    48. \(\left\{ \begin{array} {l} 2x−5y=7 \\ 3x−y=17 \end{array} \right.\)

    49. \(\left\{ \begin{array} {l} 5x−3y=−1 \\ 2x−y=2 \end{array} \right.\)

    Answer

    \((7,12)\)

    50. \(\left\{ \begin{array} {l} 3x−5y=−9 \\ 5x+2y=16 \end{array} \right.\)

    51. \(\left\{ \begin{array} {l} 4x−3y=3 \\ 2x+5y=−31 \end{array} \right.\)

    Answer

    \((−3,−5)\)

    52. \(\left\{ \begin{array} {l} 3x+8y=−3 \\ 2x+5y=−3 \end{array} \right.\)

    53. \(\left\{ \begin{array} {l} 11x+9y=−5 \\ 7x+5y=−1 \end{array} \right.\)

    Answer

    \((2,−3)\)

    54. \(\left\{ \begin{array} {l} 3x+8y=67 \\ 5x+3y=60 \end{array} \right.\)

    55. \(\left\{ \begin{array} {l} 2x+9y=−4 \\ 3x+13y=−7 \end{array} \right.\)

    Answer

    \((−11,2)\)

    56. \(\left\{ \begin{array} {l} \frac{1}{3}x−y=−3 \\ x+\frac{5}{2}y=2 \end{array} \right.\)

    57. \(\left\{ \begin{array} {l} x+\frac{1}{2}y=\frac{3}{2} \\ \frac{1}{5}x−\frac{1}{5}y=3 \end{array} \right.\)

    Answer

    \((6/−9,24/7)\)

    58. \(\left\{ \begin{array} {l} x+\frac{1}{3}y=−1 \\ \frac{1}{3}x+\frac{1}{2}y=1 \end{array} \right.\)

    59. \(\left\{ \begin{array} {l} \frac{1}{3}x−y=−3 \\ \frac{2}{3}x+\frac{5}{2}y=3 \end{array} \right.\)

    Answer

    \((−3,2)\)

    60. \(\left\{ \begin{array} {l} 2x+y=3 \\ 6x+3y=9 \end{array} \right.\)

    61. \(\left\{ \begin{array} {l} x−4y=−1 \\ −3x+12y=3 \end{array} \right.\)

    Answer

    infinitely many solutions with solution set: \(\big\{ (x,y) | x−4y=−1 \big\}\)

    62. \(\left\{ \begin{array} {l} −3x−y=8 \\ 6x+2y=−16 \end{array} \right.\)

    63. \(\left\{ \begin{array} {l} 4x+3y=2 \\ 20x+15y=10 \end{array} \right.\)

    Answer

    infinitely many solutions with solution set: \(\big\{ (x,y) | 4x+3y=2 \big\}\)

    Choose the Most Convenient Method to Solve a System of Linear Equations

    In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

    64.
    ⓐ \(\left\{ \begin{array} {l} 8x−15y=−32 \\ 6x+3y=−5 \end{array} \right.\)

    ⓑ \(\left\{ \begin{array} {l} x=4y−3 \\ 4x−2y=−6 \end{array} \right.\)

    65.
    ⓐ \(\left\{ \begin{array} {l} y=7x−5 \\ 3x−2y=16 \end{array} \right.\)

    ⓑ \(\left\{ \begin{array} {l} 12x−5y=−42 \\ 3x+7y=−15 \end{array} \right.\)

    Answer

    ⓐ substitution ⓑ elimination

    66.
    ⓐ \(\left\{ \begin{array} {l} y=4x+95 \\ x−2y=−21 \end{array} \right.\)

    ⓑ \(\left\{ \begin{array} {l} 9x−4y=24 \\ 3x+5y=−14 \end{array} \right.\)

    67.
    ⓐ \(\left\{ \begin{array} {l} 14x−15y=−30 \\ 7x+2y=10 \end{array} \right.\)

    ⓑ \(\left\{ \begin{array} {l} x=9y−11 \\ 2x−7y=−27 \end{array} \right.\)

    Answer

    ⓐ elimination ⓑ substitution

    Writing Exercises

    68. In a system of linear equations, the two equations have the same intercepts. Describe the possible solutions to the system.

    69. Solve the system of equations by substitution and explain all your steps in words: \(\left\{ \begin{array} {l} 3x+y=1 \\ 2x=y−8 \end{array} \right. \)

    Answer

    Answers will vary.

    70. Solve the system of equations by elimination and explain all your steps in words: \(\left\{ \begin{array} {l} 5x+4y=10 \\ 2x=3y+27 \end{array} \right. \)

    71. Solve the system of equations \(\left\{ \begin{array} {l} x+y=10 \\ x−y=6 \end{array} \right.\)

    ⓐ by graphing ⓑ by substitution
    ⓒ Which method do you prefer? Why?

    Answer

    Answers will vary.

    Self Check

    After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This table has 4 columns 5 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don’t get it. The first column has the following statements: determine whether an ordered pair is a solution of a system of equations, solve a system of linear equations by graphing, solve a system of equations by substitution, solve a system of equations by elimination, choose the most convenient method to solve a system of linear equations. The remaining columns are blank.

    If most of your checks were:

    …confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

    …with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

    …no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.