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9.11.1: Review Exercises

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    116441
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    Review Exercises

    Systems of Linear Equations: Two Variables

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

    1.

    3xy=4 x+4y=3 3xy=4 x+4y=3 and (1,1) (1,1)

    2.

    6x2y=24 3x+3y=18 6x2y=24 3x+3y=18 and (9,15) (9,15)

    For the following exercises, use substitution to solve the system of equations.

    3.

    10x+5y=−5 3x2y=−12 10x+5y=−5 3x2y=−12

    4.

    4 7 x+ 1 5 y= 43 70 5 6 x 1 3 y= 2 3 4 7 x+ 1 5 y= 43 70 5 6 x 1 3 y= 2 3

    5.

    5x+6y=14 4x+8y=8 5x+6y=14 4x+8y=8

    For the following exercises, use addition to solve the system of equations.

    6.

    3x+2y=−7 2x+4y=6 3x+2y=−7 2x+4y=6

    7.

    3x+4y=2 9x+12y=3 3x+4y=2 9x+12y=3

    8.

    8x+4y=2 6x5y=0.7 8x+4y=2 6x5y=0.7

    For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

    9.

    A factory has a cost of production C(x)=150x+15,000 C(x)=150x+15,000 and a revenue function R(x)=200x. R(x)=200x. What is the break-even point?

    10.

    A performer charges C(x)=50x+10,000, C(x)=50x+10,000, where x x is the total number of attendees at a show. The venue charges $75 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?

    Systems of Linear Equations: Three Variables

    For the following exercises, solve the system of three equations using substitution or addition.

    11.

    0.5x0.5y=10 0.2y+0.2x=4 0.1x+0.1z=2 0.5x0.5y=10 0.2y+0.2x=4 0.1x+0.1z=2

    12.

    5x+3yz=5 3x2y+4z=13 4x+3y+5z=22 5x+3yz=5 3x2y+4z=13 4x+3y+5z=22

    13.

    x+y+z=1 2x+2y+2z=1 3x+3y=2 x+y+z=1 2x+2y+2z=1 3x+3y=2

    14.

    2x3y+z=−1 x+y+z=−4 4x+2y3z=33 2x3y+z=−1 x+y+z=−4 4x+2y3z=33

    15.

    3x+2yz=−10 xy+2z=7 x+3y+z=−2 3x+2yz=−10 xy+2z=7 x+3y+z=−2

    16.

    3x+4z=−11 x2y=5 4yz=−10 3x+4z=−11 x2y=5 4yz=−10

    17.

    2x3y+z=0 2x+4y3z=0 6x2yz=0 2x3y+z=0 2x+4y3z=0 6x2yz=0

    18.

    6x4y2z=2 3x+2y5z=4 6y7z=5 6x4y2z=2 3x+2y5z=4 6y7z=5

    For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

    19.

    Three odd numbers sum up to 61. The smaller is one-third the larger and the middle number is 16 less than the larger. What are the three numbers?

    20.

    A local theatre sells out for their show. They sell all 500 tickets for a total purse of $8,070.00. The tickets were priced at $15 for students, $12 for children, and $18 for adults. If the band sold three times as many adult tickets as children’s tickets, how many of each type was sold?

    Systems of Nonlinear Equations and Inequalities: Two Variables

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

    21.

    y= x 2 7 y=5x13 y= x 2 7 y=5x13

    22.

    y= x 2 4 y=5x+10 y= x 2 4 y=5x+10

    23.

    x 2 + y 2 =16 y=x8 x 2 + y 2 =16 y=x8

    24.

    x 2 + y 2 =25 y= x 2 +5 x 2 + y 2 =25 y= x 2 +5

    25.

    x 2 + y 2 =4 y x 2 =3 x 2 + y 2 =4 y x 2 =3

    For the following exercises, graph the inequality.

    26.

    y> x 2 1 y> x 2 1

    27.

    1 4 x 2 + y 2 <4 1 4 x 2 + y 2 <4

    For the following exercises, graph the system of inequalities.

    28.

    x 2 + y 2 +2x<3 y> x 2 3 x 2 + y 2 +2x<3 y> x 2 3

    29.

    x 2 2x+ y 2 4x<4 y<x+4 x 2 2x+ y 2 4x<4 y<x+4

    30.

    x 2 + y 2 <1 y 2 <x x 2 + y 2 <1 y 2 <x

    Partial Fractions

    For the following exercises, decompose into partial fractions.

    31.

    2x+6 x 2 +3x+2 2x+6 x 2 +3x+2

    32.

    10x+2 4 x 2 +4x+1 10x+2 4 x 2 +4x+1

    33.

    7x+20 x 2 +10x+25 7x+20 x 2 +10x+25

    34.

    x18 x 2 12x+36 x18 x 2 12x+36

    35.

    x 2 +36x+70 x 3 125 x 2 +36x+70 x 3 125

    36.

    5 x 2 +6x2 x 3 +27 5 x 2 +6x2 x 3 +27

    37.

    x 3 4 x 2 +3x+11 ( x 2 2) 2 x 3 4 x 2 +3x+11 ( x 2 2) 2

    38.

    4 x 4 2 x 3 +22 x 2 6x+48 x ( x 2 +4) 2 4 x 4 2 x 3 +22 x 2 6x+48 x ( x 2 +4) 2

    Matrices and Matrix Operations

    For the following exercises, perform the requested operations on the given matrices.

    A=[ 4 2 1 3 ],B=[ 6 7 3 11 2 4 ],C=[ 6 7 11 2 14 0 ],D=[ 1 4 9 10 5 7 2 8 5 ],E=[ 7 14 3 2 1 3 0 1 9 ] A=[ 4 2 1 3 ],B=[ 6 7 3 11 2 4 ],C=[ 6 7 11 2 14 0 ],D=[ 1 4 9 10 5 7 2 8 5 ],E=[ 7 14 3 2 1 3 0 1 9 ]

    39.

    4A 4A

    40.

    10D6E 10D6E

    41.

    B+C B+C

    42.

    AB AB

    43.

    BA BA

    44.

    BC BC

    45.

    CB CB

    46.

    DE DE

    47.

    ED ED

    48.

    EC EC

    49.

    CE CE

    50.

    A 3 A 3

    Solving Systems with Gaussian Elimination

    For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.

    51.

    [ 1 0 −3 0 1 2 0 0 0 | 7 −5 0 ] [ 1 0 −3 0 1 2 0 0 0 | 7 −5 0 ]

    52.

    [ 1 0 5 0 1 −2 0 0 0 | −9 4 3 ] [ 1 0 5 0 1 −2 0 0 0 | −9 4 3 ]

    For the following exercises, write the augmented matrix from the system of linear equations.

    53.

    2x+2y+z=7 2x8y+5z=0 19x10y+22z=3 2x+2y+z=7 2x8y+5z=0 19x10y+22z=3

    54.

    4x+2y3z=14 12x+3y+z=100 9x6y+2z=31 4x+2y3z=14 12x+3y+z=100 9x6y+2z=31

    55.

    x+3z=12 x+4y=0 y+2z=7 x+3z=12 x+4y=0 y+2z=7

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

    56.

    3x4y=7 6x+8y=14 3x4y=7 6x+8y=14

    57.

    3x4y=1 6x+8y=6 3x4y=1 6x+8y=6

    58.

    1.1x2.3y=6.2 5.2x4.1y=4.3 1.1x2.3y=6.2 5.2x4.1y=4.3

    59.

    2x+3y+2z=1 4x6y4z=2 10x+15y+10z=0 2x+3y+2z=1 4x6y4z=2 10x+15y+10z=0

    60.

    x+2y4z=8 3y+8z=4 7x+y+2z=1 x+2y4z=8 3y+8z=4 7x+y+2z=1

    Solving Systems with Inverses

    For the following exercises, find the inverse of the matrix.

    61.

    [ 0.2 1.4 1.2 0.4 ] [ 0.2 1.4 1.2 0.4 ]

    62.

    [ 1 2 1 2 1 4 3 4 ] [ 1 2 1 2 1 4 3 4 ]

    63.

    [ 12 9 6 1 3 2 4 3 2 ] [ 12 9 6 1 3 2 4 3 2 ]

    64.

    [ 2 1 3 1 2 3 3 2 1 ] [ 2 1 3 1 2 3 3 2 1 ]

    For the following exercises, find the solutions by computing the inverse of the matrix.

    65.

    0.3x0.1y=10 0.1x+0.3y=14 0.3x0.1y=10 0.1x+0.3y=14

    66.

    0.4x0.2y=0.6 0.1x+0.05y=0.3 0.4x0.2y=0.6 0.1x+0.05y=0.3

    67.

    4x+3y3z=4.3 5x4yz=6.1 x+z=0.7 4x+3y3z=4.3 5x4yz=6.1 x+z=0.7

    68.

    2x3y+2z=3 x+2y+4z=5 2y+5z=3 2x3y+2z=3 x+2y+4z=5 2y+5z=3

    For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

    69.

    Students were asked to bring their favorite fruit to class. 90% of the fruits consisted of banana, apple, and oranges. If oranges were half as popular as bananas and apples were 5% more popular than bananas, what are the percentages of each individual fruit?

    70.

    A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at $2 and the chocolate chip cookies at $1. They raised $250 and sold 175 items. How many brownies and how many cookies were sold?

    Solving Systems with Cramer's Rule

    For the following exercises, find the determinant.

    71.

    | 100 0 0 0 | | 100 0 0 0 |

    72.

    | 0.2 0.6 0.7 1.1 | | 0.2 0.6 0.7 1.1 |

    73.

    | 1 4 3 0 2 3 0 0 3 | | 1 4 3 0 2 3 0 0 3 |

    74.

    | 2 0 0 0 2 0 0 0 2 | | 2 0 0 0 2 0 0 0 2 |

    For the following exercises, use Cramer’s Rule to solve the linear systems of equations.

    75.

    4x2y=23 5x10y=35 4x2y=23 5x10y=35

    76.

    0.2x0.1y=0 0.3x+0.3y=2.5 0.2x0.1y=0 0.3x+0.3y=2.5

    77.

    0.5x+0.1y=0.3 0.25x+0.05y=0.15 0.5x+0.1y=0.3 0.25x+0.05y=0.15

    78.

    x+6y+3z=4 2x+y+2z=3 3x2y+z=0 x+6y+3z=4 2x+y+2z=3 3x2y+z=0

    79.

    4x3y+5z= 5 2 7x9y3z= 3 2 x5y5z= 5 2 4x3y+5z= 5 2 7x9y3z= 3 2 x5y5z= 5 2

    80.

    3 10 x 1 5 y 3 10 z= 1 50 1 10 x 1 10 y 1 2 z= 9 50 2 5 x 1 2 y 3 5 z= 1 5 3 10 x 1 5 y 3 10 z= 1 50 1 10 x 1 10 y 1 2 z= 9 50 2 5 x 1 2 y 3 5 z= 1 5


    9.11.1: Review Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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