Exam 6: Systems of Equations and Inequalities

Use the properties of determinants to find the value of the second determinant, given the value of the first. - $\left| \begin{array} { c c c } x & y & z \\u & v & w \\1 & - 3 & 4\end{array} \right| = - 138 \left| \begin{array} { c c c } x & y & z \\u & v & w \\2 & - 6 & 8\end{array} \right| = ?$ x y z a v w 1 -3 4 vertical barequalsminus138vertical bar x y z u v w 2 -6 8 \midequalsquestion mark

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Question 1

Correct Answer:

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A

Which method should be used to solve the system? Explain your answer, including a description of the first step. -7x² - 3y² = 36 5x + 5y = -5

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Question 2

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Substitution should be used, because using elimination would mean introducing an xy term into the
second equation unnecessarily. The second equation should be solved for x or y.

Which method should be used to solve the system? Explain your answer, including a description of the first step. -x² + 9y² = 9 7x² + 8y² = 81

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Question 3

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Either method may be used. Elimination may be used since the equations have the same types of terms (
x² , y2, constant). If elimination is used, then the first equation should be multiplied by 7 or -7.
Substitution may be used efficiently, since a coefficient of the first equation is 1. If substitution is used,
then the first equation should be solved for x² .

Find the inverse of the matrix. - $A = \left[ \begin{array} { r r } 14 & - 6 \\- 7 & 3\end{array} \right]$

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Question 4

Find the inverse of the matrix. - $\left[ \begin{array} { r r } 0 & -6\\-4 & 5 \\\end{array} \right]$

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Question 5

Use a graphing utility to solve the system of equations. Express the solution rounded to two decimal places. - $\left\{ \begin{array} { r } x ^ { 3 } + y ^ { 2 } = 2 \\x ^ { 2 } y = 4\end{array} \right.$

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Question 6

Solve the system of equations using substitution. - $\left\{ \begin{array} { l } \ln x = 3 \ln y \\ 3 ^ { x } = 27 y \\ \end{array} \right.$

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Question 7

Find the inverse of the matrix. - $\left[ \begin{array} { c c c } 2 & - 5 & 3 \\2 & - 4 & 8 \\4 & - 9 & 11\end{array} \right]$

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Question 8

Which method should be used to solve the system? Explain your answer, including a description of the first step. -x² - y² = 16 x - 8y = -6

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Question 9

Solve the problem. -A retired couple has $140,000 to invest to obtain annual income. They want some of it invested in safe Certificates of Deposit yielding 5%. The rest they want to invest in AA bonds yielding 12% per year. How much Should they invest in each to realize exactly $13,300 per year?

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Question 10

Tell whether the given rational expression is proper or improper. - $\frac { x ^ { 2 } - 12 x + 36 } { \left( x ^ { 2 } - 8 x + 15 \right) ( x + 3 ) }$

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Question 11

Use the properties of determinants to find the value of the second determinant, given the value of the first. - $\left| \begin{array} { c c c } x & y & z \\u & v & w \\1 & - 3 & - 4\end{array} \right| = 62 \left| \begin{array} { c c c } 1 & - 3 & - 4 \\3 u & 3 v & 3 w \\x - 1 & y + 3 & z + 4\end{array} \right| = ?$

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Question 12

Solve the system of equations using substitution. - $\left\{ \begin{array} { l } x y - x ^ { 2 } = - 20 \\x - 2 y = 3\end{array} \right.$

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Question 13

Solve the system of equations by using substitution. - $\left\{ \begin{array} { c } x + 6 y = 6 \\4 x - 9 y = - 9\end{array} \right.$

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Question 14

Write the partial fraction decomposition of the rational expression. - $\frac { 2 x - 5 } { x ^ { 2 } - 5 x - 6 }$

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Question 15

Graph the system of inequalities. -

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Question 16

Solve the problem. -The Family Arts Center charges $21 for adults, $13 for senior citizens, and $8 for children under 12 for their live performances on Sunday afternoon. This past Sunday, the paid revenue was $12,388 for 905 tickets sold. There Were 50 more children than adults. How many children attended?

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Question 17

Use a graphing utility to find the inverse of the matrix, if it exists. Round answers to two decimal places, if necessary. - $\left[ \begin{array} { r r r r } 1 & - 4 & 0 & 0 \\0 & 1 & - 2 & 0 \\0 & 0 & 1 & - 1 \\0 & 0 & 0 & 1\end{array} \right]$

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Question 18

Solve the system of equations using substitution. - $\left\{ \begin{array} { r } x y = 20 \\x + y = 9\end{array} \right.$

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Question 19

Graph the inequality. -y ≤-3

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Question 20

Which method should be used to solve the system? Explain your answer, including a description of the first step. -7x² - 3y² = 36 5x + 5y = -5

Which method should be used to solve the system? Explain your answer, including a description of the first step. -x² + 9y² = 9 7x² + 8y² = 81

Find the inverse of the matrix. - $A = \left[ \begin{array} { r r } 14 & - 6 \\- 7 & 3\end{array} \right]$

Find the inverse of the matrix. - $\left[ \begin{array} { r r } 0 & -6\\-4 & 5 \\\end{array} \right]$

Use a graphing utility to solve the system of equations. Express the solution rounded to two decimal places. - $\left\{ \begin{array} { r } x ^ { 3 } + y ^ { 2 } = 2 \\x ^ { 2 } y = 4\end{array} \right.$

Solve the system of equations using substitution. - $\left\{ \begin{array} { l } \ln x = 3 \ln y \\ 3 ^ { x } = 27 y \\ \end{array} \right.$

Find the inverse of the matrix. - $\left[ \begin{array} { c c c } 2 & - 5 & 3 \\2 & - 4 & 8 \\4 & - 9 & 11\end{array} \right]$

Which method should be used to solve the system? Explain your answer, including a description of the first step. -x² - y² = 16 x - 8y = -6

Solve the problem. -A retired couple has $140,000 to invest to obtain annual income. They want some of it invested in safe Certificates of Deposit yielding 5%. The rest they want to invest in AA bonds yielding 12% per year. How much Should they invest in each to realize exactly $13,300 per year?

Tell whether the given rational expression is proper or improper. - $\frac { x ^ { 2 } - 12 x + 36 } { \left( x ^ { 2 } - 8 x + 15 \right) ( x + 3 ) }$

Solve the system of equations using substitution. - $\left\{ \begin{array} { l } x y - x ^ { 2 } = - 20 \\x - 2 y = 3\end{array} \right.$

Solve the system of equations by using substitution. - $\left\{ \begin{array} { c } x + 6 y = 6 \\4 x - 9 y = - 9\end{array} \right.$

Write the partial fraction decomposition of the rational expression. - $\frac { 2 x - 5 } { x ^ { 2 } - 5 x - 6 }$

Graph the system of inequalities. -

Use a graphing utility to find the inverse of the matrix, if it exists. Round answers to two decimal places, if necessary. - $\left[ \begin{array} { r r r r } 1 & - 4 & 0 & 0 \\0 & 1 & - 2 & 0 \\0 & 0 & 1 & - 1 \\0 & 0 & 0 & 1\end{array} \right]$

Solve the system of equations using substitution. - $\left\{ \begin{array} { r } x y = 20 \\x + y = 9\end{array} \right.$

Graph the inequality. -y ≤-3

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Conics

Sequences; Induction; the Binomial Theorem

Counting and Probability

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Exam 6: Systems of Equations and Inequalities

Which method should be used to solve the system? Explain your answer, including a description of the first step. -7x2 - 3y2 = 36 5x + 5y = -5

Which method should be used to solve the system? Explain your answer, including a description of the first step. -x2 + 9y2 = 9 7x2 + 8y2 = 81

Find the inverse of the matrix. - A=[14−6−73]A = \left[ \begin{array} { r r } 14 & - 6 \\- 7 & 3\end{array} \right]A=[14−7​−63​]

Find the inverse of the matrix. - [0−6−45]\left[ \begin{array} { r r } 0 & -6\\-4 & 5 \\\end{array} \right][0−4​−65​]

Use a graphing utility to solve the system of equations. Express the solution rounded to two decimal places. - {x3+y2=2x2y=4\left\{ \begin{array} { r } x ^ { 3 } + y ^ { 2 } = 2 \\x ^ { 2 } y = 4\end{array} \right.{x3+y2=2x2y=4​

Solve the system of equations using substitution. - {ln⁡x=3ln⁡y3x=27y\left\{ \begin{array} { l } \ln x = 3 \ln y \\ 3 ^ { x } = 27 y \\ \end{array} \right.{lnx=3lny3x=27y​

Find the inverse of the matrix. - [2−532−484−911]\left[ \begin{array} { c c c } 2 & - 5 & 3 \\2 & - 4 & 8 \\4 & - 9 & 11\end{array} \right]​224​−5−4−9​3811​​

Which method should be used to solve the system? Explain your answer, including a description of the first step. -x2 - y2 = 16 x - 8y = -6

Solve the problem. -A retired couple has $140,000 to invest to obtain annual income. They want some of it invested in safe Certificates of Deposit yielding 5%. The rest they want to invest in AA bonds yielding 12% per year. How much Should they invest in each to realize exactly $13,300 per year?

Tell whether the given rational expression is proper or improper. - x2−12x+36(x2−8x+15)(x+3)\frac { x ^ { 2 } - 12 x + 36 } { \left( x ^ { 2 } - 8 x + 15 \right) ( x + 3 ) }(x2−8x+15)(x+3)x2−12x+36​

Solve the system of equations using substitution. - {xy−x2=−20x−2y=3\left\{ \begin{array} { l } x y - x ^ { 2 } = - 20 \\x - 2 y = 3\end{array} \right.{xy−x2=−20x−2y=3​

Solve the system of equations by using substitution. - {x+6y=64x−9y=−9\left\{ \begin{array} { c } x + 6 y = 6 \\4 x - 9 y = - 9\end{array} \right.{x+6y=64x−9y=−9​

Write the partial fraction decomposition of the rational expression. - 2x−5x2−5x−6\frac { 2 x - 5 } { x ^ { 2 } - 5 x - 6 }x2−5x−62x−5​

Graph the system of inequalities. -

Solve the system of equations using substitution. - {xy=20x+y=9\left\{ \begin{array} { r } x y = 20 \\x + y = 9\end{array} \right.{xy=20x+y=9​

Graph the inequality. -y ≤-3

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Conics

Sequences; Induction; the Binomial Theorem

Counting and Probability

Filters

Which method should be used to solve the system? Explain your answer, including a description of the first step. -7x² - 3y² = 36 5x + 5y = -5

Which method should be used to solve the system? Explain your answer, including a description of the first step. -x² + 9y² = 9 7x² + 8y² = 81

Find the inverse of the matrix. - $A = \left[ \begin{array} { r r } 14 & - 6 \\- 7 & 3\end{array} \right]$

Find the inverse of the matrix. - $\left[ \begin{array} { r r } 0 & -6\\-4 & 5 \\\end{array} \right]$

Use a graphing utility to solve the system of equations. Express the solution rounded to two decimal places. - $\left\{ \begin{array} { r } x ^ { 3 } + y ^ { 2 } = 2 \\x ^ { 2 } y = 4\end{array} \right.$

Solve the system of equations using substitution. - $\left\{ \begin{array} { l } \ln x = 3 \ln y \\ 3 ^ { x } = 27 y \\ \end{array} \right.$

Find the inverse of the matrix. - $\left[ \begin{array} { c c c } 2 & - 5 & 3 \\2 & - 4 & 8 \\4 & - 9 & 11\end{array} \right]$

Which method should be used to solve the system? Explain your answer, including a description of the first step. -x² - y² = 16 x - 8y = -6

Tell whether the given rational expression is proper or improper. - $\frac { x ^ { 2 } - 12 x + 36 } { \left( x ^ { 2 } - 8 x + 15 \right) ( x + 3 ) }$

Solve the system of equations using substitution. - $\left\{ \begin{array} { l } x y - x ^ { 2 } = - 20 \\x - 2 y = 3\end{array} \right.$

Solve the system of equations by using substitution. - $\left\{ \begin{array} { c } x + 6 y = 6 \\4 x - 9 y = - 9\end{array} \right.$

Write the partial fraction decomposition of the rational expression. - $\frac { 2 x - 5 } { x ^ { 2 } - 5 x - 6 }$

Solve the system of equations using substitution. - $\left\{ \begin{array} { r } x y = 20 \\x + y = 9\end{array} \right.$