Solve the system by using Cramer's rule. $\begin{aligned} & 7 x+2 y=-4 \\ & 4 x+3 y=7 \end{aligned}$

The answer of Solve the system by using Cramer's rule. \(\begin{aligned} &...

Exam 7: Analytic Geometry and Nonlinear Systems

Consider the system of equations x-3y+2z =5 2x-4y+3z =12 3x-7y+6z =23 (a) Write the matrix of coefficients $A$ , the matrix of variables $X$ , and the matrix of constants $B$ for this system. (b) Find $A^{-1}$ . (c) Use the matrix inverse method to solve the system. (d) If the matrix of constants $B$ is replaced by the matrix $\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ , find the solution to $A X=B$ .

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Question 1

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(a) $A=\left[\begin{array}{lll}1 & -3 & 2 \\ 2 & -4 & 3 \\ 3 & -7 & 6\end{array}\right] ; X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right] ; B=\left[\begin{array}{r}5 \\ 12 \\ 23\end{array}\right]$
(b) $A^{-1}=\left[\begin{array}{rrr}-\frac{3}{2} & 2 & -\frac{1}{2} \\ -\frac{3}{2} & 0 & \frac{1}{2} \\ -1 & -1 & 1\end{array}\right]$
(c) $(5,4,6)$
(d) $(0,0,0)$

Suppose that $A$ is a $1 \times n$ matrix and $B$ is a $n \times 1$ matrix. (a) Can $A B$ be found? If so, give the dimensions of the resulting matrix. (b) Can $B A$ be found? If so, give the dimensions of the resulting matrix. (c) Does $A B=B A$ ? Explain why or why not. (d) If $C$ and $D$ are each square matrices, will $C D$ always exist?

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Question 2

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(a) Yes; $1 \times 1$
(b) Yes; $n \times n$
(c) No; They have different dimensions.
(d) No; $C$ and $D$ must have the same dimensions.

Find the partial fraction decomposition for the rational expression $\frac{x^{2}+9 x-18}{(x+2)(x-2)^{2}}$ .

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Question 3

Correct Answer:

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$\frac{3}{x-2}-\frac{2}{x+2}+\frac{1}{(x-2)^{2}}$

Evaluate each determinant. (a) $\operatorname{det}\left[\begin{array}{rr}-6 & 9 \\ 4 & -5\end{array}\right]$ (b) $\operatorname{det}\left[\begin{array}{rrr}2 & 0 & -2 \\ 3 & 1 & 5 \\ 6 & 4 & 3\end{array}\right]$

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Question 4

The solution set of a system of inequalities is shown below. Which of the following systems is it? $The solution set of a system of inequalities is shown below. Which of the following systems is it? (a) y+x \leq 7 y \geq x^{2}-4 x+3 (b) y+x \geq 7 y \geq x^{2}-4 x+3 (c) \begin{aligned} & y+x \geq 7 \\ & y \leq x^{2}-4 x+3\end{aligned} (d) y+x \leq 7 y \leq x^{2}-4 x+3$ (a) $y+x \leq 7$ $y \geq x^{2}-4 x+3$ (b) $y+x \geq 7$ $y \geq x^{2}-4 x+3$ (c) y+x\geq7 y\leq-4x+3 (d) $y+x \leq 7$ $y \leq x^{2}-4 x+3$

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Question 5

Find the partial fraction decomposition for the rational expression $\frac{4 x+7}{x^{2}-9 x+14}$ .

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Question 6

Evaluate each determinant. (a) $\operatorname{det}\left[\begin{array}{rr}3 & 11 \\ -2 & 8\end{array}\right]$ (b) $\operatorname{det}\left[\begin{array}{rrr}3 & 2 & -2 \\ 1 & -3 & 4 \\ 2 & 5 & 1\end{array}\right]$

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Question 7

Evaluate each determinant. (a) $\operatorname{det}\left[\begin{array}{rr}8 & 6 \\ -2 & -7\end{array}\right]$ (b) $\operatorname{det}\left[\begin{array}{rrr}1 & 3 & 4 \\ -2 & 4 & 5 \\ 7 & 6 & -3\end{array}\right]$

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Question 8

Solve the system by using Cramer's rule. 7x+2y=-4 4x+3y=7

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Question 9

Find the partial fraction decomposition for the rational expression $\frac{-2 x^{2}-2 x+48}{(x+3)(x-3)^{2}}$ .

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Question 10

Consider the system of equations 3+4y =12 -9x+2y =-18 (a) What type of graph does each equation have? (b) How many points of intersection are possible for these types of graphs? (c) Solve the system. (d) Support the solutions using a graphing calculator.

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Question 11

The minimum yearly temperature in a certain temperate climate was recorded each year for five years beginning in 2000. The minimum temperature increased for the first few years before declining. The table shows the temperature, in degrees Fahrenheit, for those years, with 2000 represented by year 0, 2001 by year 1, and so on. Use the data points $(0,38.8),(2,47.3)$ , and $(4,39)$ to find a quadratic function defined by $f(x)=a x^{2}+b x+c$ that models the data. Graph $f$ together with the data. $The minimum yearly temperature in a certain temperate climate was recorded each year for five years beginning in 2000. The minimum temperature increased for the first few years before declining. The table shows the temperature, in degrees Fahrenheit, for those years, with 2000 represented by year 0, 2001 by year 1, and so on. Use the data points (0,38.8),(2,47.3) , and (4,39) to find a quadratic function defined by f(x)=a x^{2}+b x+c that models the data. Graph f together with the data.$

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Question 12

Perform the following matrix operations if possible. (a) $\left[\begin{array}{rr}1 & -2 \\ 4 & 7 \\ -3 & 0\end{array}\right]-\left[\begin{array}{rr}0 & 7 \\ 5 & 3 \\ -5 & 8\end{array}\right]$ (b) $\left[\begin{array}{rrr}6 & 2 & -3 \\ 7 & -9 & 10\end{array}\right]+\left[\begin{array}{rr}5 & -4 \\ 3 & 8\end{array}\right]$ (c) $\left[\begin{array}{rr}2 & -1 \\ 0 & 3 \\ 1 & 5\end{array}\right]\left[\begin{array}{rrr}5 & -2 & 3 \\ -3 & 4 & -1\end{array}\right]$

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Question 13

Consider the system of equations $x+2 y-3 z=8$ $2 x+5 y-11 z=18$ $3 x+7 y-13 z=24$ (a) Write the matrix of coefficients $A$ , the matrix of variables $X$ , and the matrix of constants $B$ for this system. (b) Find $A^{-1}$ . (c) Use the matrix inverse method to solve the system. (d) If the matrix of constants $B$ is replaced by the matrix $\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ , find the solution to $A X=B$ .

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Question 14

The employment rate in a city has been documented at two year intervals. The table shows the number of employed citizens, in millions, tor several years, with 2000 represented by year 0,2002 by year 2 , and so on. Use the data points $(0,2.9),(4,2.3)$ , and $(8,3.0)$ to find a quadratic function defined by $f(x)=a x^{2}+b x+c$ that models the data. Graph $f$ together with the data. $The employment rate in a city has been documented at two year intervals. The table shows the number of employed citizens, in millions, tor several years, with 2000 represented by year 0,2002 by year 2 , and so on. Use the data points (0,2.9),(4,2.3) , and (8,3.0) to find a quadratic function defined by f(x)=a x^{2}+b x+c that models the data. Graph f together with the data.$

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Question 15

Perform the following matrix operations if possible. (a) $\left[\begin{array}{rr}3 & -1 \\ 5 & 2\end{array}\right]\left[\begin{array}{rrr}1 & -4 & 1 \\ 3 & 2 & -2\end{array}\right]$ (b) $\left[\begin{array}{rrr}2 & -2 & 6 \\ 1 & 0 & -7\end{array}\right]+3\left[\begin{array}{rrr}-1 & 1 & -2 \\ 1 & 3 & 4\end{array}\right]$ (c) $\left[\begin{array}{rrr}3 & -4 & 5 \\ 7 & 10 & -4\end{array}\right]\left[\begin{array}{rrr}5 & 2 & -9 \\ 0 & 1 & 5\end{array}\right]$

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Question 16

Perform the following matrix operations if possible. (a) $\left[\begin{array}{rr}5 & -3 \\ 7 & -2 \\ 0 & 4\end{array}\right]\left[\begin{array}{rr}1 & 2 \\ -1 & 6 \\ 3 & 3\end{array}\right]$ (b) $\left[\begin{array}{rr}1 & -3 \\ 2 & 0 \\ 4 & -5\end{array}\right]-3\left[\begin{array}{rr}-1 & 4 \\ 3 & 4 \\ 2 & -2\end{array}\right]$ (c) $\left[\begin{array}{rr}2 & 0 \\ -3 & -1 \\ 4 & 6\end{array}\right]\left[\begin{array}{rrr}1 & 2 & 3 \\ -1 & 0 & -2\end{array}\right]$

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Question 17

Consider the system of equations $3 x^{2}-4 y=12$ $-9 x+2 y=-18$ (a) What type of graph does each equation have? (b) How many points of intersection are possible for these types of graphs? (c) Solve the system. (d) Support the solutions using a graphing calculator.

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Question 18

Solve the system by using Cramer's rule. $5 x+9 y=2$ $7 x-18 y=-4$

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Question 19

Suppose that $A$ is a $m \times 3$ matrix and $B$ is a $4 \times m$ matrix. (a) Can $A B$ be found? If so, give the dimensions of the resulting matrix. (b) Can $B A$ be found? If so, give the dimensions of the resulting matrix. (c) Does $A B=B A$ ? Explain why or why not. (d) If $C$ and $D$ are each square matrices, will $C D$ always exist?

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Question 20

Find the partial fraction decomposition for the rational expression $\frac{x^{2}+9 x-18}{(x+2)(x-2)^{2}}$ .

Evaluate each determinant. (a) $\operatorname{det}\left[\begin{array}{rr}-6 & 9 \\ 4 & -5\end{array}\right]$ (b) $\operatorname{det}\left[\begin{array}{rrr}2 & 0 & -2 \\ 3 & 1 & 5 \\ 6 & 4 & 3\end{array}\right]$

Find the partial fraction decomposition for the rational expression $\frac{4 x+7}{x^{2}-9 x+14}$ .

Evaluate each determinant. (a) $\operatorname{det}\left[\begin{array}{rr}3 & 11 \\ -2 & 8\end{array}\right]$ (b) $\operatorname{det}\left[\begin{array}{rrr}3 & 2 & -2 \\ 1 & -3 & 4 \\ 2 & 5 & 1\end{array}\right]$

Evaluate each determinant. (a) $\operatorname{det}\left[\begin{array}{rr}8 & 6 \\ -2 & -7\end{array}\right]$ (b) $\operatorname{det}\left[\begin{array}{rrr}1 & 3 & 4 \\ -2 & 4 & 5 \\ 7 & 6 & -3\end{array}\right]$

Solve the system by using Cramer's rule. 7x+2y=-4 4x+3y=7

Find the partial fraction decomposition for the rational expression $\frac{-2 x^{2}-2 x+48}{(x+3)(x-3)^{2}}$ .

Consider the system of equations 3+4y =12 -9x+2y =-18 (a) What type of graph does each equation have? (b) How many points of intersection are possible for these types of graphs? (c) Solve the system. (d) Support the solutions using a graphing calculator.

Consider the system of equations $3 x^{2}-4 y=12$ $-9 x+2 y=-18$ (a) What type of graph does each equation have? (b) How many points of intersection are possible for these types of graphs? (c) Solve the system. (d) Support the solutions using a graphing calculator.

Solve the system by using Cramer's rule. $5 x+9 y=2$ $7 x-18 y=-4$

Linear Functions, Equations, and Inequalities

Analysis of Graphs of Functions

Polynomial Functions

Rational, Power, and Root Functions

Inverse, Exponential, and Logarithmic Functions

Systems and Matrices

The Unit Circle and Functions of Trigonometry

Trigonometric Identities and Equations

Applications of Trigonometry and Vectors

Further Topics in Algebra

Limits, Derivatives, and Definite Integrals

Reference: Basic Algebraic Concepts

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Exam 7: Analytic Geometry and Nonlinear Systems

Find the partial fraction decomposition for the rational expression x2+9x−18(x+2)(x−2)2\frac{x^{2}+9 x-18}{(x+2)(x-2)^{2}}(x+2)(x−2)2x2+9x−18​ .

Evaluate each determinant. (a) det⁡[−694−5]\operatorname{det}\left[\begin{array}{rr}-6 & 9 \\ 4 & -5\end{array}\right]det[−64​9−5​] (b) det⁡[20−2315643]\operatorname{det}\left[\begin{array}{rrr}2 & 0 & -2 \\ 3 & 1 & 5 \\ 6 & 4 & 3\end{array}\right]det​236​014​−253​​

Find the partial fraction decomposition for the rational expression 4x+7x2−9x+14\frac{4 x+7}{x^{2}-9 x+14}x2−9x+144x+7​ .

Evaluate each determinant. (a) det⁡[311−28]\operatorname{det}\left[\begin{array}{rr}3 & 11 \\ -2 & 8\end{array}\right]det[3−2​118​] (b) det⁡[32−21−34251]\operatorname{det}\left[\begin{array}{rrr}3 & 2 & -2 \\ 1 & -3 & 4 \\ 2 & 5 & 1\end{array}\right]det​312​2−35​−241​​

Solve the system by using Cramer's rule. 7x+2y=-4 4x+3y=7

Find the partial fraction decomposition for the rational expression −2x2−2x+48(x+3)(x−3)2\frac{-2 x^{2}-2 x+48}{(x+3)(x-3)^{2}}(x+3)(x−3)2−2x2−2x+48​ .

Consider the system of equations 3+4y =12 -9x+2y =-18 (a) What type of graph does each equation have? (b) How many points of intersection are possible for these types of graphs? (c) Solve the system. (d) Support the solutions using a graphing calculator.

Solve the system by using Cramer's rule. 5x+9y=25 x+9 y=25x+9y=2 7x−18y=−47 x-18 y=-47x−18y=−4

Linear Functions, Equations, and Inequalities

Analysis of Graphs of Functions

Polynomial Functions

Rational, Power, and Root Functions

Inverse, Exponential, and Logarithmic Functions

Systems and Matrices

The Unit Circle and Functions of Trigonometry

Trigonometric Identities and Equations

Applications of Trigonometry and Vectors

Further Topics in Algebra

Limits, Derivatives, and Definite Integrals

Reference: Basic Algebraic Concepts

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Find the partial fraction decomposition for the rational expression $\frac{x^{2}+9 x-18}{(x+2)(x-2)^{2}}$ .

Evaluate each determinant. (a) $\operatorname{det}\left[\begin{array}{rr}-6 & 9 \\ 4 & -5\end{array}\right]$ (b) $\operatorname{det}\left[\begin{array}{rrr}2 & 0 & -2 \\ 3 & 1 & 5 \\ 6 & 4 & 3\end{array}\right]$

Find the partial fraction decomposition for the rational expression $\frac{4 x+7}{x^{2}-9 x+14}$ .

Evaluate each determinant. (a) $\operatorname{det}\left[\begin{array}{rr}3 & 11 \\ -2 & 8\end{array}\right]$ (b) $\operatorname{det}\left[\begin{array}{rrr}3 & 2 & -2 \\ 1 & -3 & 4 \\ 2 & 5 & 1\end{array}\right]$

Find the partial fraction decomposition for the rational expression $\frac{-2 x^{2}-2 x+48}{(x+3)(x-3)^{2}}$ .

Solve the system by using Cramer's rule. $5 x+9 y=2$ $7 x-18 y=-4$