Solve the system of equations below using Gaussian elimination. \[\begin{array} { c } 5 x + 4 y + 3 z = - 72 \\ x - 2 y + 2 z = - 48 \\ x - y - z = 24 \end{array}\]

A) $( 1 , - 1 , - 22 )$ B) $( 1,0 , - 23 )$ C) $( 0,1 , - 25 )$ D) $( - 1,1 , - 26 )$ E) $( 0,0 , - 24 )$ A) $( 1 , - 1 , - 22 )$ B) $( 1,0 , - 23 )$ C) $( 0,1 , - 25 )$ D) $( - 1,1 , - 26 )$ E) $( 0,0 , - 24 )$

Solve the system of equations below using Gaussian elimination. $\begin{aligned} 5 x + 4 y + 3 z & = 54 \\ x - 2 y + 2 z & = 36 \\ x - y - z & = - 18 \end{aligned}$

A) $( 1 , - 1,20 )$ B) $( 1,0,19 )$ C) $( 0,1,17 )$ D) $( - 1,1,16 )$ E) $( 0,0,18 )$ A) $( 1 , - 1,20 )$ B) $( 1,0,19 )$ C) $( 0,1,17 )$ D) $( - 1,1,16 )$ E) $( 0,0,18 )$

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. $\begin{array} { r } \left\{ \begin{aligned} x + 5 y - 5 z & = 42 \\ y - 9 z & = 48 \\ z & = - 5 \end{aligned} \right. \\ \left\{ \begin{aligned} x + 8 y - 3 z & = 31 \\ y + 5 z & = - 22 \\ z & = - 5 \end{aligned} \right. \end{array}$

A) $x = 2 , y = 3 , z = - 5$ B) $x = - 2 , y = - 3 , z = - 5$ C) $x = - 3 , y = - 5 , z = 2$ D) $x = 3 , y = - 5 , z = 2$ E) The systems yield different solutions. A) $x = 2 , y = 3 , z = - 5$ B) $x = - 2 , y = - 3 , z = - 5$ C) $x = - 3 , y = - 5 , z = 2$ D) $x = 3 , y = - 5 , z = 2$ E) The systems yield different solutions.

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. $\begin{array} { l } \left\{ \begin{aligned} x + y - 9 z & = - 67 \\ y + 5 z & = 31 \\ z & = 7 \end{aligned} \right. \\ \left\{ \begin{aligned} x - 9 y + 6 z & = 76 \\ y - 9 z & = - 67 \\ z & = 7 \end{aligned} \right. \end{array}$

A) $x = 0 , y = - 4 , z = 7$ B) $x = 0 , y = 4 , z = 7$ C) $x = 4 , y = 7 , z = 0$ D) $x = - 4 , y = 7 , z = 0$ E) The systems yield different solutions. A) $x = 0 , y = - 4 , z = 7$ B) $x = 0 , y = 4 , z = 7$ C) $x = 4 , y = 7 , z = 0$ D) $x = - 4 , y = 7 , z = 0$ E) The systems yield different solutions.

Fill in the blank using elementary row operations to form a row-equivalent matrix. \[\begin{array} { l } { \left[ \begin{array} { c c c } 2 & - 1 & - 9 \\ 8 & - 6 & 4 \end{array} \right] } \\ { \left[ \begin{array} { c c c } 2 & - 1 & - 9 \\ 0 & \square & 40 \end{array} \right] } \end{array}\]

A) $\left[ \begin{array} { r r r } 2 & - 1 & - 9 \\ 0 & - 2 & 40 \end{array} \right]$ B) $\left[ \begin{array} { c c c } 2 & - 1 & - 9 \\ 0 & 0 & 40 \end{array} \right]$ C) $\left[ \begin{array} { c c c } 2 & - 1 & - 9 \\ 0 & - 10 & 40 \end{array} \right]$ D) $\left[ \begin{array} { c c c } 2 & - 1 & - 9 \\ 0 & - 1 & 40 \end{array} \right]$ E) $\left[ \begin{array} { c c c } 2 & - 1 & - 9 \\ 0 & 2 & 40 \end{array} \right]$ A) $\left[ \begin{array} { r r r } 2 & - 1 & - 9 \\ 0 & - 2 & 40 \end{array} \right]$ B) $\left[ \begin{array} { c c c } 2 & - 1 & - 9 \\ 0 & 0 & 40 \end{array} \right]$ C) $\left[ \begin{array} { c c c } 2 & - 1 & - 9 \\ 0 & - 10 & 40 \end{array} \right]$ D) $\left[ \begin{array} { c c c } 2 & - 1 & - 9 \\ 0 & - 1 & 40 \end{array} \right]$ E) $\left[ \begin{array} { c c c } 2 & - 1 & - 9 \\ 0 & 2 & 40 \end{array} \right]$

Exam 7: Linear Systems and Matrices

An augmented matrix that represents a system of linear equations (in variables $x , y$ , and $z$ ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $\left[ \begin{array} { c c c : c } 1 & 0 & 0 & - 1 \\0 & 1 & 0 & 3 \\0 & 0 & 1 & 8\end{array} \right]$

(Multiple Choice)

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Question 81

Use a determinant to determine whether the points below are collinear. $( 3 , - 1 ) , ( 0 , - 3 ) , ( 27,15 )$

(Multiple Choice)

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Question 82

Solve system by the method of substitution and graph your solution. $\left\{ \begin{array} { l } y = x ^ { 3 } + 3 x ^ { 2 } + 2 x \\y = - x\end{array} \right.$

(Multiple Choice)

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Question 83

Solve the system of equations algebraically. $\left\{ \begin{array} { r } x y + 25 = 0 \\49 x - 100 y - 700 = 0\end{array} \right.$

(Multiple Choice)

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Question 84

If possible, find $5 A + 4 B$ . $A = \left[ \begin{array} { c c c } - 8 & 9 & 2 \\ - 2 & 1 & - 4 \end{array} \right] , B = \left[ \begin{array} { c c c } 8 & 2 & 1 \\ - 3 & 8 & 4 \end{array} \right]$

(Multiple Choice)

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Question 85

Find $| A B |$ , if $A = \left[ \begin{array} { r r r } 5 & - 2 & - 2 \\- 3 & 4 & - 1 \\- 5 & 6 & 2\end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 1 & - 3 \\- 5 & - 3 & 4 \\- 5 & - 1 & - 2\end{array} \right]$

(Multiple Choice)

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Question 86

An augmented matrix that represents a system of linear equations (in variables $x$ , $y$ , and $z$ ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $\left[ \begin{array} { c c c : c } 1 & 0 & 0 & 9 \\0 & 1 & 0 & 8 \\0 & 0 & 1 & - 7\end{array} \right]$

(Multiple Choice)

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Question 87

Solve the system of equations below using Gaussian elimination. 5x+4y+3z=-72 x-2y+2z=-48 x-y-z=24

(Multiple Choice)

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Question 88

Solve system of equations by the method of substitution. $\left\{ \begin{aligned}12 x ^ { 3 } - 9 y & = 0 \\12 x - y & = 0\end{aligned} \right.$

(Multiple Choice)

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Question 89

Solve the system of linear equations. $\left\{ \begin{aligned}x + 5 y + 5 z & = 9 \\- 2 x + z & = - 6 \\2 x + 4 y - 3 z & = - 2\end{aligned} \right.$

(Multiple Choice)

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Question 90

Find the inverse of the matrix $\left[ \begin{array} { c c } 4 & - 5 \\ 1 & 5 \end{array} \right]$ (if it exists).

(Multiple Choice)

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Question 91

Evaluate the expression. $\frac { 1 } { 4 } \left[ \begin{array} { l l l } - 9 & 1 & - 1\end{array} \right] + \left[ \begin{array} { l l l } - 4 & - 6 & 8\end{array} \right]$

(Multiple Choice)

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Question 92

Solve the system of equations below using Gaussian elimination. 5x+4y+3z =54 x-2y+2z =36 x-y-z =-18

(Multiple Choice)

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Question 93

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. x+5y-5z =42 y-9z =48 z =-5 x+8y-3z =31 y+5z =-22 z =-5

(Multiple Choice)

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Question 94

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. x+y-9z =-67 y+5z =31 z =7 x-9y+6z =76 y-9z =-67 z =7

(Multiple Choice)

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Question 95

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $\frac { x + 9 } { x \left( x ^ { 2 } + 3 \right) ^ { 2 } }$

(Multiple Choice)

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Question 96

Fill in the blank using elementary row operations to form a row-equivalent matrix. 2 -1 -9 8 -6 4 2 -1 -9 0 \square 40

(Multiple Choice)

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Question 97

Solve the system by the method of elimination. Round numbers to three decimal places. $\left\{ \begin{array} { c } 5.7 x + 7.1 y = 3.2 \\- 0.8 x - 4.8 y = - 1.1\end{array} \right.$

(Multiple Choice)

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Question 98

Perform the indicated row operations on the matrix. Show the final result. $\left[ \begin{array} { c c c } 1 & 7 & 1 \\- 7 & - 48 & - 8 \\- 5 & - 1 & - 39\end{array} \right]$ Add 7 times $R _ { 1 }$ to $R _ { 2 }$ . Add 5 times $R _ { 1 }$ to $R _ { 3 }$ .

(Multiple Choice)

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Question 99

If possible, find $3 A + 4 B$ . $A = \left[ \begin{array} { c c c } - 4 & 3 & 9 \\3 & - 2 & 5\end{array} \right] , B = \left[ \begin{array} { c c c } - 8 & 9 & 5 \\0 & 2 & 1\end{array} \right]$

(Multiple Choice)

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Question 100

Use a determinant to determine whether the points below are collinear. $( 3 , - 1 ) , ( 0 , - 3 ) , ( 27,15 )$

Solve system by the method of substitution and graph your solution. $\left\{ \begin{array} { l } y = x ^ { 3 } + 3 x ^ { 2 } + 2 x \\y = - x\end{array} \right.$

Solve the system of equations algebraically. $\left\{ \begin{array} { r } x y + 25 = 0 \\49 x - 100 y - 700 = 0\end{array} \right.$

If possible, find $5 A + 4 B$ . $A = \left[ \begin{array} { c c c } - 8 & 9 & 2 \\ - 2 & 1 & - 4 \end{array} \right] , B = \left[ \begin{array} { c c c } 8 & 2 & 1 \\ - 3 & 8 & 4 \end{array} \right]$

Find $| A B |$ , if $A = \left[ \begin{array} { r r r } 5 & - 2 & - 2 \\- 3 & 4 & - 1 \\- 5 & 6 & 2\end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 1 & - 3 \\- 5 & - 3 & 4 \\- 5 & - 1 & - 2\end{array} \right]$

Solve the system of equations below using Gaussian elimination. 5x+4y+3z=-72 x-2y+2z=-48 x-y-z=24

Solve system of equations by the method of substitution. $\left\{ \begin{aligned}12 x ^ { 3 } - 9 y & = 0 \\12 x - y & = 0\end{aligned} \right.$

Solve the system of linear equations. $\left\{ \begin{aligned}x + 5 y + 5 z & = 9 \\- 2 x + z & = - 6 \\2 x + 4 y - 3 z & = - 2\end{aligned} \right.$

Find the inverse of the matrix $\left[ \begin{array} { c c } 4 & - 5 \\ 1 & 5 \end{array} \right]$ (if it exists).

Evaluate the expression. $\frac { 1 } { 4 } \left[ \begin{array} { l l l } - 9 & 1 & - 1\end{array} \right] + \left[ \begin{array} { l l l } - 4 & - 6 & 8\end{array} \right]$

Solve the system of equations below using Gaussian elimination. 5x+4y+3z =54 x-2y+2z =36 x-y-z =-18

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. x+5y-5z =42 y-9z =48 z =-5 x+8y-3z =31 y+5z =-22 z =-5

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. x+y-9z =-67 y+5z =31 z =7 x-9y+6z =76 y-9z =-67 z =7

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $\frac { x + 9 } { x \left( x ^ { 2 } + 3 \right) ^ { 2 } }$

Fill in the blank using elementary row operations to form a row-equivalent matrix. 2 -1 -9 8 -6 4 2 -1 -9 0 \square 40

Solve the system by the method of elimination. Round numbers to three decimal places. $\left\{ \begin{array} { c } 5.7 x + 7.1 y = 3.2 \\- 0.8 x - 4.8 y = - 1.1\end{array} \right.$

Perform the indicated row operations on the matrix. Show the final result. $\left[ \begin{array} { c c c } 1 & 7 & 1 \\- 7 & - 48 & - 8 \\- 5 & - 1 & - 39\end{array} \right]$ Add 7 times $R _ { 1 }$ to $R _ { 2 }$ . Add 5 times $R _ { 1 }$ to $R _ { 3 }$ .

If possible, find $3 A + 4 B$ . $A = \left[ \begin{array} { c c c } - 4 & 3 & 9 \\3 & - 2 & 5\end{array} \right] , B = \left[ \begin{array} { c c c } - 8 & 9 & 5 \\0 & 2 & 1\end{array} \right]$

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Exam 7: Linear Systems and Matrices

Use a determinant to determine whether the points below are collinear. (3,−1),(0,−3),(27,15)( 3 , - 1 ) , ( 0 , - 3 ) , ( 27,15 )(3,−1),(0,−3),(27,15)

Solve system by the method of substitution and graph your solution. {y=x3+3x2+2xy=−x\left\{ \begin{array} { l } y = x ^ { 3 } + 3 x ^ { 2 } + 2 x \\y = - x\end{array} \right.{y=x3+3x2+2xy=−x​

Solve the system of equations algebraically. {xy+25=049x−100y−700=0\left\{ \begin{array} { r } x y + 25 = 0 \\49 x - 100 y - 700 = 0\end{array} \right.{xy+25=049x−100y−700=0​

If possible, find 5A+4B5 A + 4 B5A+4B . A=[−892−21−4],B=[821−384]A = \left[ \begin{array} { c c c } - 8 & 9 & 2 \\ - 2 & 1 & - 4 \end{array} \right] , B = \left[ \begin{array} { c c c } 8 & 2 & 1 \\ - 3 & 8 & 4 \end{array} \right]A=[−8−2​91​2−4​],B=[8−3​28​14​]

Solve the system of equations below using Gaussian elimination. 5x+4y+3z=-72 x-2y+2z=-48 x-y-z=24

Solve system of equations by the method of substitution. {12x3−9y=012x−y=0\left\{ \begin{aligned}12 x ^ { 3 } - 9 y & = 0 \\12 x - y & = 0\end{aligned} \right.{12x3−9y12x−y​=0=0​

Solve the system of linear equations. {x+5y+5z=9−2x+z=−62x+4y−3z=−2\left\{ \begin{aligned}x + 5 y + 5 z & = 9 \\- 2 x + z & = - 6 \\2 x + 4 y - 3 z & = - 2\end{aligned} \right.⎩⎨⎧​x+5y+5z−2x+z2x+4y−3z​=9=−6=−2​

Find the inverse of the matrix [4−515]\left[ \begin{array} { c c } 4 & - 5 \\ 1 & 5 \end{array} \right][41​−55​] (if it exists).

Evaluate the expression. 14[−91−1]+[−4−68]\frac { 1 } { 4 } \left[ \begin{array} { l l l } - 9 & 1 & - 1\end{array} \right] + \left[ \begin{array} { l l l } - 4 & - 6 & 8\end{array} \right]41​[−9​1​−1​]+[−4​−6​8​]

Solve the system of equations below using Gaussian elimination. 5x+4y+3z =54 x-2y+2z =36 x-y-z =-18

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. x+5y-5z =42 y-9z =48 z =-5 x+8y-3z =31 y+5z =-22 z =-5

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. x+y-9z =-67 y+5z =31 z =7 x-9y+6z =76 y-9z =-67 z =7

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. x+9x(x2+3)2\frac { x + 9 } { x \left( x ^ { 2 } + 3 \right) ^ { 2 } }x(x2+3)2x+9​

Fill in the blank using elementary row operations to form a row-equivalent matrix. 2 -1 -9 8 -6 4 2 -1 -9 0 \square 40

Solve the system by the method of elimination. Round numbers to three decimal places. {5.7x+7.1y=3.2−0.8x−4.8y=−1.1\left\{ \begin{array} { c } 5.7 x + 7.1 y = 3.2 \\- 0.8 x - 4.8 y = - 1.1\end{array} \right.{5.7x+7.1y=3.2−0.8x−4.8y=−1.1​

If possible, find 3A+4B3 A + 4 B3A+4B . A=[−4393−25],B=[−895021]A = \left[ \begin{array} { c c c } - 4 & 3 & 9 \\3 & - 2 & 5\end{array} \right] , B = \left[ \begin{array} { c c c } - 8 & 9 & 5 \\0 & 2 & 1\end{array} \right]A=[−43​3−2​95​],B=[−80​92​51​]

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

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Limits and an Introduction to Calculus

Filters

Use a determinant to determine whether the points below are collinear. $( 3 , - 1 ) , ( 0 , - 3 ) , ( 27,15 )$

Solve system by the method of substitution and graph your solution. $\left\{ \begin{array} { l } y = x ^ { 3 } + 3 x ^ { 2 } + 2 x \\y = - x\end{array} \right.$

Solve the system of equations algebraically. $\left\{ \begin{array} { r } x y + 25 = 0 \\49 x - 100 y - 700 = 0\end{array} \right.$

If possible, find $5 A + 4 B$ . $A = \left[ \begin{array} { c c c } - 8 & 9 & 2 \\ - 2 & 1 & - 4 \end{array} \right] , B = \left[ \begin{array} { c c c } 8 & 2 & 1 \\ - 3 & 8 & 4 \end{array} \right]$

Solve system of equations by the method of substitution. $\left\{ \begin{aligned}12 x ^ { 3 } - 9 y & = 0 \\12 x - y & = 0\end{aligned} \right.$

Solve the system of linear equations. $\left\{ \begin{aligned}x + 5 y + 5 z & = 9 \\- 2 x + z & = - 6 \\2 x + 4 y - 3 z & = - 2\end{aligned} \right.$

Find the inverse of the matrix $\left[ \begin{array} { c c } 4 & - 5 \\ 1 & 5 \end{array} \right]$ (if it exists).

Evaluate the expression. $\frac { 1 } { 4 } \left[ \begin{array} { l l l } - 9 & 1 & - 1\end{array} \right] + \left[ \begin{array} { l l l } - 4 & - 6 & 8\end{array} \right]$

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $\frac { x + 9 } { x \left( x ^ { 2 } + 3 \right) ^ { 2 } }$

Solve the system by the method of elimination. Round numbers to three decimal places. $\left\{ \begin{array} { c } 5.7 x + 7.1 y = 3.2 \\- 0.8 x - 4.8 y = - 1.1\end{array} \right.$

If possible, find $3 A + 4 B$ . $A = \left[ \begin{array} { c c c } - 4 & 3 & 9 \\3 & - 2 & 5\end{array} \right] , B = \left[ \begin{array} { c c c } - 8 & 9 & 5 \\0 & 2 & 1\end{array} \right]$