Question 1

Use matrices to solve the system of linear equations below. 
 {−2x−y=−13x+2y=11\left\{ \begin{array} { l } - 2 x - y = - 13 \x + 2 y = 11\end{array} ight.{−2x−y=−13x+2y=11​

Accepted Answer

A)   (-2,2) 
B)   (3,-1) 
C)   (5,3) 
D)   (5,-2) 
E)   (-1,4)

Question 2

Solve the system of linear equations below by the method of elimination.  {4r−5s=52r−5s=15\left\{ \begin{array} { c } 4 r - 5 s = 5 \2 r - 5 s = 15\end{array} ight.{4r−5s=52r−5s=15​

Accepted Answer

A)    r=9 and s=−9r = 9 	ext { and } s = - 9r=9 and s=−9 
B)    r=−1 and s=−3r = - 1 	ext { and } s = - 3r=−1 and s=−3 
C)    r=−8 and s=−8r = - 8 	ext { and } s = - 8r=−8 and s=−8 
D)    r=−5 and s=−5r = - 5 	ext { and } s = - 5r=−5 and s=−5 
E)    r=2 and s=4r = 2 	ext { and } s = 4r=2 and s=4

Question 3

Solve for x in the matrix below by using elementary row operations to form a row-equivalent matrix. 
 [63−20−2−3−3−940−38]\left[ \begin{array} { c c c c } 6 & 3 & - 2 & 0 \- 2 & - 3 & - 3 & - 9 \4 & 0 & - 3 & 8\end{array} ight]​6−24​3−30​−2−3−3​0−98​​  
 [63−20−2−3−3−9−2−3xy]\left[ \begin{array} { c c c c } 6 & 3 & - 2 & 0 \- 2 & - 3 & - 3 & - 9 \- 2 & - 3 & x & y\end{array} ight]​6−2−2​3−3−3​−2−3x​0−9y​​

Accepted Answer

A)   -6
B)   -1
C)   -15
D)   15
E)   10

Question 4

A total of  $15,000\$ 15,000$15,000  is invested in two bonds that pay  5.5%5.5 \%5.5%  and  6%6 \%6%  simple interest. The annual interest is  $865\$ 865$865  . How much is invested in the  6%6 \%6%  bond?

Accepted Answer

A)    $7,000\$ 7,000$7,000 
B)    $4,000\$ 4,000$4,000 
C)    $3,000\$ 3,000$3,000 
D)    $9,000\$ 9,000$9,000 
E)    $8,000\$ 8,000$8,000

Question 5

The sum of three positive numbers is  656565  . The second number is 5   less 	ext { less } less   than the first, and the third is 5 times the first. What is the   third 	ext { third } third   number?

Accepted Answer

A)    101010 
B)    505050 
C)    121212 
D)    202020 
E)    242424

Question 6

Use the graph of the equation  {8x−5y=143x+5y=8\left\{ \begin{array} { l } 8 x - 5 y = 14 \3 x + 5 y = 8\end{array} ight.{8x−5y=143x+5y=8​  to determine whether the system has any solutions. Find any solutions that exist.

Accepted Answer

A)    (2,25) \left( 2 , \frac { 2 } { 5 } \right) (2,52​)  
B)    (25,2) \left( \frac { 2 } { 5 } , 2 \right) (52​,2)  
C)    (2,2) ( 2,2 ) (2,2)  
D)  infinitely many solutions
E)  no solution

Question 7

Describe the elementary row operation used to transform the first matrix  [3560220−3−4]\left[ \begin{array} { c c c } 3 & 5 & 6 \0 & 2 & 2 \0 & - 3 & - 4\end{array} ight]​300​52−3​62−4​​  into the second matrix  [356022032]\left[ \begin{array} { l l l } 3 & 5 & 6 \0 & 2 & 2 \0 & 3 & 2\end{array} ight]​300​523​622​​  .

Accepted Answer

A)  Add 4 times the third row to the second row.
B)  Add 4 times the second row to the third row.
C)  Add 3 times the third row to the second row.
D)  Add 3 times the second row to the third row.
E)  Add 2 times the second row to the third row.

Question 8

Form the coefficient matrix for the system of linear equations below. 
 {6x+3y+7z=0x+6y+z=78x+9y+7z=−9\left\{ \begin{array} { r } 6 x + 3 y + 7 z = 0 \x + 6 y + z = 7 \8 x + 9 y + 7 z = - 9\end{array} ight.⎩⎨⎧​6x+3y+7z=0x+6y+z=78x+9y+7z=−9​

Accepted Answer

A)    [63701617897−9]\left[ \begin{array} { c c c c } 6 & 3 & 7 & 0 \1 & 6 & 1 & 7 \8 & 9 & 7 & - 9\end{array} ight]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z'/>​618​369​717​07−9​M347 1759 V0 H263 V1759 v0 v1759 h84z'/>​ 
B)    [637161897]\left[ \begin{array} { l l l } 6 & 3 & 7 \1 & 6 & 1 \8 & 9 & 7\end{array} ight]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z'/>​618​369​717​M347 1759 V0 H263 V1759 v0 v1759 h84z'/>​ 
C)    [61836971707−9]\left[ \begin{array} { c c c } 6 & 1 & 8 \3 & 6 & 9 \7 & 1 & 7 \0 & 7 & - 9\end{array} ight]H403z M403 1759 V0 H319 V1759 v1200 v1759 h84z'/>​6370​1617​897−9​M347 1759 V0 H263 V1759 v1200 v1759 h84z'/>​ 
D)    [618369717]\left[ \begin{array} { l l l } 6 & 1 & 8 \3 & 6 & 9 \7 & 1 & 7\end{array} ight]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z'/>​637​161​897​M347 1759 V0 H263 V1759 v0 v1759 h84z'/>​ 
E)    [07−9]\left[ \begin{array} { c } 0 \7 \- 9\end{array} ight]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z'/>​07−9​M347 1759 V0 H263 V1759 v0 v1759 h84z'/>​

Question 9

Determine whether the system is consistent or inconsistent.  {x+y=12x+2y=6 \left\{\begin{array}{l} x+y=1 \ 2 x+2 y=6\end{array}ight. {x+y=12x+2y=6​

Accepted Answer

A)  The system is consistent.
B)  The system is inconsistent.

Question 10

Evaluate the determinant of the matrix. Expand by minors along the row or column that appears to make the computation easiest. 
 [1041−42−123]\left[ \begin{array} { c c c } 1 & 0 & 4 \1 & - 4 & 2 \- 1 & 2 & 3\end{array} ight]​11−1​0−42​423​​

Accepted Answer

A)   -5
B)   11
C)   8
D)   -24
E)   -4

Question 11

Solve the system of linear equations below.  {4x+3y+z=65x−4y−2z=−42x+5y−3z=38\left\{ \begin{array} { c } 4 x + 3 y + z = 6 \5 x - 4 y - 2 z = - 4 \2 x + 5 y - 3 z = 38\end{array} ight.⎩⎨⎧​4x+3y+z=65x−4y−2z=−42x+5y−3z=38​

Accepted Answer

A)    (4,−6,0) ( 4 , - 6,0 ) (4,−6,0)  
B)    (0,−6,4) ( 0 , - 6,4 ) (0,−6,4)  
C)    (0,4,−6) ( 0,4 , - 6 ) (0,4,−6)  
D)    (−6,4,0) ( - 6,4,0 ) (−6,4,0)  
E)    (−6,0,4) ( - 6,0,4 ) (−6,0,4)

Question 12

Solve the system of linear equations below.  {x+y−2z=−8x−y−3z=−52x+4z=−4\left\{ \begin{array} { r } x + y - 2 z = - 8 \x - y - 3 z = - 5 \2 x + 4 z = - 4\end{array} ight.⎩⎨⎧​x+y−2z=−8x−y−3z=−52x+4z=−4​

Accepted Answer

A)     4 \x + 6 y \leq 9\end{array} ight.">{3x−y>4x+6y≤9\left\{ \begin{array} { l } 3 x - y > 4 \x + 6 y \leq 9\end{array} ight.{3x−y>4x+6y≤9​ 
B)    (−4,−2,1) ( - 4 , - 2,1 ) (−4,−2,1)  
C)    (−4,1,−2) ( - 4,1 , - 2 ) (−4,1,−2)  
D)    (−2,−4,1) ( - 2 , - 4,1 ) (−2,−4,1)  
E)    (−2,1,−4) ( - 2,1 , - 4 ) (−2,1,−4)

Question 13

Solve the system of equations below by the method of substitution.  {9x+5y=−665x+9y=−74\left\{ \begin{array} { l } 9 x + 5 y = - 66 \5 x + 9 y = - 74\end{array} ight.{9x+5y=−665x+9y=−74​

Accepted Answer

A)   (2,-5) 
B)   (-2,-8) 
C)   (-4,-6) 
D)   (0,-5) 
E)    (5,-5)

Question 14

A grocer wishes to mix three kinds of nuts to obtain  404040  pounds of a mixture priced at  $4.45\$ 4.45$4.45  per pound. Peanuts cost  $3\$ 3$3  per pound, almonds cost  $5\$ 5$5  per pound, and pistachios cost  $5.5\$ 5.5$5.5  per pound. Half of the mixture is composed of peanuts and almonds. How many pounds of   peanuts 	ext { peanuts } peanuts   should the grocer use?

Accepted Answer

A)    161616  pounds
B)    202020  pounds
C)    282828  pounds
D)    151515  pounds
E)    303030  pounds

Question 15

Solve the system of linear equations below by the method of elimination, if a single solution exists.  {−x+2y=3−2x+4y=8\left\{ \begin{array} { c } - x + 2 y = 3 \- 2 x + 4 y = 8\end{array} ight.{−x+2y=3−2x+4y=8​

Accepted Answer

A)    x=9 and y=−8x = 9 	ext { and } y = - 8x=9 and y=−8 
B)    x=−5 and y=−1x = - 5 	ext { and } y = - 1x=−5 and y=−1 
C)    x=2 and y=6x = 2 	ext { and } y = 6x=2 and y=6 
D)    x=−4 and y=8x = - 4 	ext { and } y = 8x=−4 and y=8 
E)  Infinitely many solutions

Question 16

Graph the system of linear inequalities below.  {x+y≥−2x−y≤−1\left\{ \begin{array} { l } x + y \geq - 2 \x - y \leq - 1\end{array} ight.{x+y≥−2x−y≤−1​

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 17

A person plans to invest up to $10,000 in two different interest-bearing accounts, account X and account Y. Account Y is to contain at least $  100010001000  . Moreover, account X should have at least twice the amount in account Y. Graph the system of linear inequalities describing the various amounts that can be deposited in each account.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 18

Find two positive integers that satisfy the requirements that the difference of four times the smaller number and the larger number is 14 and the sum of the smaller number and four times the larger number is 335 .

Accepted Answer

A)   23 and 110
B)   23 and 78
C)   31 and  787878 
D)    313131  and  110110110 
E)    313131  and  626262

Question 19

Solve the system of linear equations below by the method of elimination, if a single solution exists.  {−4x+y=−412x−3y=3\left\{ \begin{array} { l } - 4 x + y = - 4 \12 x - 3 y = 3\end{array} ight.{−4x+y=−412x−3y=3​

Accepted Answer

A)    x=8 and y=−8x = 8 	ext { and } y = - 8x=8 and y=−8 
B)    x=1 and y=0x = 1 	ext { and } y = 0x=1 and y=0 
C)    x=9 and y=4x = 9 	ext { and } y = 4x=9 and y=4 
D)    x=−4 and y=0x = - 4 	ext { and } y = 0x=−4 and y=0 
E)  no solution

Question 20

Determine the value of k such that the system of linear equations below is inconsistent.  {7x+18y=−35x+ky=3\left\{ \begin{array} { c } 7 x + 18 y = - 3 \5 x + k y = 3\end{array} ight.{7x+18y=−35x+ky=3​

Accepted Answer

A)    k=−607k = - \frac { 60 } { 7 }k=−760​ 
B)    k=−556k = - \frac { 55 } { 6 }k=−655​ 
C)    k=907k = \frac { 90 } { 7 }k=790​ 
D)    k=994k = \frac { 99 } { 4 }k=499​ 
E)    k=409k = \frac { 40 } { 9 }k=940​

Exam 4: Systems of Equations and Inequalities

Correct Answer
verified

Use matrices to solve the system of linear equations below. $\left\{ \begin{array} { l } - 2 x - y = - 13 \\x + 2 y = 11\end{array} \right.$

Correct Answer
verified

Solve the system of linear equations below by the method of elimination, if a single solution exists. $\left\{ \begin{array} { l } - 4 x + y = - 4 \\12 x - 3 y = 3\end{array} \right.$

Correct Answer
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A total of $\$ 15,000$ is invested in two bonds that pay $5.5 \%$ and $6 \%$ simple interest. The annual interest is $\$ 865$ . How much is invested in the $6 \%$ bond?

Correct Answer
verified

Determine whether the system is consistent or inconsistent. $\left\{\begin{array}{l} x+y=1 \\ 2 x+2 y=6\end{array}\right.$

Correct Answer
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Describe the elementary row operation used to transform the first matrix $\left[ \begin{array} { c c c } 3 & 5 & 6 \\0 & 2 & 2 \\0 & - 3 & - 4\end{array} \right]$ into the second matrix $\left[ \begin{array} { l l l } 3 & 5 & 6 \\0 & 2 & 2 \\0 & 3 & 2\end{array} \right]$ .

Correct Answer
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Find two positive integers that satisfy the requirements that the difference of four times the smaller number and the larger number is 14 and the sum of the smaller number and four times the larger number is 335 .

Correct Answer
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Solve the system of linear equations below by the method of elimination, if a single solution exists. $\left\{ \begin{array} { c } - x + 2 y = 3 \\- 2 x + 4 y = 8\end{array} \right.$

Correct Answer
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Evaluate the determinant of the matrix. Expand by minors along the row or column that appears to make the computation easiest. $\left[ \begin{array} { c c c } 1 & 0 & 4 \\1 & - 4 & 2 \\- 1 & 2 & 3\end{array} \right]$

Correct Answer
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Correct Answer
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Correct Answer
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Solve the system of equations below by the method of substitution. $\left\{ \begin{array} { l } 9 x + 5 y = - 66 \\5 x + 9 y = - 74\end{array} \right.$

Correct Answer
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Form the coefficient matrix for the system of linear equations below. $\left\{ \begin{array} { r } 6 x + 3 y + 7 z = 0 \\x + 6 y + z = 7 \\8 x + 9 y + 7 z = - 9\end{array} \right.$

Correct Answer
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Solve the system of linear equations below by the method of elimination. $\left\{ \begin{array} { c } 4 r - 5 s = 5 \\2 r - 5 s = 15\end{array} \right.$

Correct Answer
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Determine the value of k such that the system of linear equations below is inconsistent. $\left\{ \begin{array} { c } 7 x + 18 y = - 3 \\5 x + k y = 3\end{array} \right.$

Correct Answer
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Correct Answer
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Graph the system of linear inequalities below. $\left\{ \begin{array} { l } x + y \geq - 2 \\x - y \leq - 1\end{array} \right.$

Correct Answer
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Solve the system of linear equations below. $\left\{ \begin{array} { c } 4 x + 3 y + z = 6 \\5 x - 4 y - 2 z = - 4 \\2 x + 5 y - 3 z = 38\end{array} \right.$

Correct Answer
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Solve the system of linear equations below. $\left\{ \begin{array} { r } x + y - 2 z = - 8 \\x - y - 3 z = - 5 \\2 x + 4 z = - 4\end{array} \right.$

Correct Answer
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The sum of three positive numbers is $65$ . The second number is 5 $\text { less }$ than the first, and the third is 5 times the first. What is the $\text { third }$ number?

Correct Answer
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Exam 4: Systems of Equations and Inequalities

Correct AnswerverifiedShow Answer

Use matrices to solve the system of linear equations below. {−2x−y=−13x+2y=11\left\{ \begin{array} { l } - 2 x - y = - 13 \\x + 2 y = 11\end{array} \right.{−2x−y=−13x+2y=11​

Correct AnswerverifiedShow Answer

Solve the system of linear equations below by the method of elimination, if a single solution exists. {−4x+y=−412x−3y=3\left\{ \begin{array} { l } - 4 x + y = - 4 \\12 x - 3 y = 3\end{array} \right.{−4x+y=−412x−3y=3​

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A total of $15,000\$ 15,000$15,000 is invested in two bonds that pay 5.5%5.5 \%5.5% and 6%6 \%6% simple interest. The annual interest is $865\$ 865$865 . How much is invested in the 6%6 \%6% bond?

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Determine whether the system is consistent or inconsistent. {x+y=12x+2y=6 \left\{\begin{array}{l} x+y=1 \\ 2 x+2 y=6\end{array}\right. {x+y=12x+2y=6​

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Correct AnswerverifiedShow Answer

Find two positive integers that satisfy the requirements that the difference of four times the smaller number and the larger number is 14 and the sum of the smaller number and four times the larger number is 335 .

Correct AnswerverifiedShow Answer

Solve the system of linear equations below by the method of elimination, if a single solution exists. {−x+2y=3−2x+4y=8\left\{ \begin{array} { c } - x + 2 y = 3 \\- 2 x + 4 y = 8\end{array} \right.{−x+2y=3−2x+4y=8​

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Evaluate the determinant of the matrix. Expand by minors along the row or column that appears to make the computation easiest. [1041−42−123]\left[ \begin{array} { c c c } 1 & 0 & 4 \\1 & - 4 & 2 \\- 1 & 2 & 3\end{array} \right]​11−1​0−42​423​​

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Correct AnswerverifiedShow Answer

Use the graph of the equation {8x−5y=143x+5y=8\left\{ \begin{array} { l } 8 x - 5 y = 14 \\3 x + 5 y = 8\end{array} \right.{8x−5y=143x+5y=8​ to determine whether the system has any solutions. Find any solutions that exist.

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Solve the system of equations below by the method of substitution. {9x+5y=−665x+9y=−74\left\{ \begin{array} { l } 9 x + 5 y = - 66 \\5 x + 9 y = - 74\end{array} \right.{9x+5y=−665x+9y=−74​

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Form the coefficient matrix for the system of linear equations below. {6x+3y+7z=0x+6y+z=78x+9y+7z=−9\left\{ \begin{array} { r } 6 x + 3 y + 7 z = 0 \\x + 6 y + z = 7 \\8 x + 9 y + 7 z = - 9\end{array} \right.⎩⎨⎧​6x+3y+7z=0x+6y+z=78x+9y+7z=−9​

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Solve the system of linear equations below by the method of elimination. {4r−5s=52r−5s=15\left\{ \begin{array} { c } 4 r - 5 s = 5 \\2 r - 5 s = 15\end{array} \right.{4r−5s=52r−5s=15​

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Determine the value of k such that the system of linear equations below is inconsistent. {7x+18y=−35x+ky=3\left\{ \begin{array} { c } 7 x + 18 y = - 3 \\5 x + k y = 3\end{array} \right.{7x+18y=−35x+ky=3​

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Correct AnswerverifiedShow Answer

Graph the system of linear inequalities below. {x+y≥−2x−y≤−1\left\{ \begin{array} { l } x + y \geq - 2 \\x - y \leq - 1\end{array} \right.{x+y≥−2x−y≤−1​

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Solve the system of linear equations below. {4x+3y+z=65x−4y−2z=−42x+5y−3z=38\left\{ \begin{array} { c } 4 x + 3 y + z = 6 \\5 x - 4 y - 2 z = - 4 \\2 x + 5 y - 3 z = 38\end{array} \right.⎩⎨⎧​4x+3y+z=65x−4y−2z=−42x+5y−3z=38​

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Solve the system of linear equations below. {x+y−2z=−8x−y−3z=−52x+4z=−4\left\{ \begin{array} { r } x + y - 2 z = - 8 \\x - y - 3 z = - 5 \\2 x + 4 z = - 4\end{array} \right.⎩⎨⎧​x+y−2z=−8x−y−3z=−52x+4z=−4​

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The sum of three positive numbers is 656565 . The second number is 5 less \text { less } less than the first, and the third is 5 times the first. What is the third \text { third } third number?

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Correct Answer
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Use matrices to solve the system of linear equations below. $\left\{ \begin{array} { l } - 2 x - y = - 13 \\x + 2 y = 11\end{array} \right.$

Correct Answer
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Solve the system of linear equations below by the method of elimination, if a single solution exists. $\left\{ \begin{array} { l } - 4 x + y = - 4 \\12 x - 3 y = 3\end{array} \right.$

Correct Answer
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A total of $\$ 15,000$ is invested in two bonds that pay $5.5 \%$ and $6 \%$ simple interest. The annual interest is $\$ 865$ . How much is invested in the $6 \%$ bond?

Correct Answer
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Determine whether the system is consistent or inconsistent. $\left\{\begin{array}{l} x+y=1 \\ 2 x+2 y=6\end{array}\right.$

Correct Answer
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Correct Answer
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Correct Answer
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Solve the system of linear equations below by the method of elimination, if a single solution exists. $\left\{ \begin{array} { c } - x + 2 y = 3 \\- 2 x + 4 y = 8\end{array} \right.$

Correct Answer
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Evaluate the determinant of the matrix. Expand by minors along the row or column that appears to make the computation easiest. $\left[ \begin{array} { c c c } 1 & 0 & 4 \\1 & - 4 & 2 \\- 1 & 2 & 3\end{array} \right]$

Correct Answer
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Correct Answer
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Correct Answer
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Solve the system of equations below by the method of substitution. $\left\{ \begin{array} { l } 9 x + 5 y = - 66 \\5 x + 9 y = - 74\end{array} \right.$

Correct Answer
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Form the coefficient matrix for the system of linear equations below. $\left\{ \begin{array} { r } 6 x + 3 y + 7 z = 0 \\x + 6 y + z = 7 \\8 x + 9 y + 7 z = - 9\end{array} \right.$

Correct Answer
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Solve the system of linear equations below by the method of elimination. $\left\{ \begin{array} { c } 4 r - 5 s = 5 \\2 r - 5 s = 15\end{array} \right.$

Correct Answer
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Determine the value of k such that the system of linear equations below is inconsistent. $\left\{ \begin{array} { c } 7 x + 18 y = - 3 \\5 x + k y = 3\end{array} \right.$

Correct Answer
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Correct Answer
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Graph the system of linear inequalities below. $\left\{ \begin{array} { l } x + y \geq - 2 \\x - y \leq - 1\end{array} \right.$

Correct Answer
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Solve the system of linear equations below. $\left\{ \begin{array} { c } 4 x + 3 y + z = 6 \\5 x - 4 y - 2 z = - 4 \\2 x + 5 y - 3 z = 38\end{array} \right.$

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Solve the system of linear equations below. $\left\{ \begin{array} { r } x + y - 2 z = - 8 \\x - y - 3 z = - 5 \\2 x + 4 z = - 4\end{array} \right.$

Correct Answer
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The sum of three positive numbers is $65$ . The second number is 5 $\text { less }$ than the first, and the third is 5 times the first. What is the $\text { third }$ number?

Correct Answer
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