Question 1

Solve the problem.
-The area of a garden is 9720 square feet, and the length of its diagonal is 153 feet. Find the dimensions of the garden.

Accepted Answer

A) 648 feet by 15 feet 
B) 72 feet by 135 feet 
C) 120 feet by 81 feet 
D) 8 feet by 1215 feet 
A) 648 feet by 15 feet 
B) 72 feet by 135 feet 
C) 120 feet by 81 feet 
D) 8 feet by 1215 feet

Question 2

Graph the system of inequalities.
-$\begin{array}{l}
y>-5 \
x \geq 3
\end{array}$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 3

Write the partial fraction decomposition of the rational expression.
-$\frac { 16 x ^ { 2 } - 105 x + 90 } { x ( x - 6 ) ( x - 5 ) }$

Accepted Answer

A)  $\frac { 3 } { x } + \frac { 7 } { x - 6 } + \frac { 6 } { x - 5 }$ 
B)  $\frac { 3 } { x } + \frac { 6 } { x - 6 } + \frac { 7 } { x - 5 }$ 
C)  $\frac { x + 6 } { x ^ { 2 } - 6 } + \frac { 7 } { x - 5 }$ 
D)  $\frac { 3 } { x } + \frac { 5 } { x - 6 } + \frac { 8 } { x - 5 }$ 
A)  $\frac { 3 } { x } + \frac { 7 } { x - 6 } + \frac { 6 } { x - 5 }$ 
B)  $\frac { 3 } { x } + \frac { 6 } { x - 6 } + \frac { 7 } { x - 5 }$ 
C)  $\frac { x + 6 } { x ^ { 2 } - 6 } + \frac { 7 } { x - 5 }$ 
D)  $\frac { 3 } { x } + \frac { 5 } { x - 6 } + \frac { 8 } { x - 5 }$

Question 4

Graph the system of inequalities.
-$\begin{array}{r}
2 x+3 y \geq 6 \
x-y \geq 3 \
y \leq 2
\end{array}$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 5

Use Gauss-Jordan elimination to solve the linear system and determine whether the system has a unique solution, no solution, or infinitely many solutions. If the system has infinitely many solutions, describe the solution as an ordered
triple involving variable z.
-$$\begin{aligned}
x + y + z & = 7 \
x - y + 2 z & = 7 \
2 x + 3 z & = 14
\end{aligned}$$

Accepted Answer

A)  $\left( - \frac { 3 z } { 2 } - 7,2 z , z \right)$ 
B)  $\left( - \frac { 3 z } { 2 } + 7,2 z , z \right)$ 
C)  $\left( - \frac { 3 z } { 2 } + 7 , \frac { z } { 2 } , z \right)$ 
D)  $\left( - \frac { 3 z } { 2 } - 7 , \frac { z } { 2 } , z \right)$ 
A)  $\left( - \frac { 3 z } { 2 } - 7,2 z , z \right)$ 
B)  $\left( - \frac { 3 z } { 2 } + 7,2 z , z \right)$ 
C)  $\left( - \frac { 3 z } { 2 } + 7 , \frac { z } { 2 } , z \right)$ 
D)  $\left( - \frac { 3 z } { 2 } - 7 , \frac { z } { 2 } , z \right)$

Question 6

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form.
-$$\begin{array} { l } 
x + y + z = 0 \
x - y + 4 z = 0 \
2 x + y + z = 5
\end{array}$$

Accepted Answer

A) (-2, -3, 5) 
B) (-2, 5, -3) 
C) (5, -3, -2) 
D) no solution 
A) (-2, -3, 5) 
B) (-2, 5, -3) 
C) (5, -3, -2) 
D) no solution

Question 7

Solve the system of equations by substitution.
-x + 7y = -2 3x + y = 34

Accepted Answer

A) (7, 12) 
B) (3, 7) 
C) (-2, 3) 
D) (12, -2) 
A) (7, 12) 
B) (3, 7) 
C) (-2, 3) 
D) (12, -2)

Question 8

Determine the real solutions to the system of nonlinear equations.
-$$\begin{array} { l } 
y = 6 x ^ { 2 } - 5 x \
y = 2 x + 3
\end{array}$$

Accepted Answer

A)  $\left( \frac { 3 } { 2 } , 6 \right) , \left( - \frac { 1 } { 3 } , - \frac { 7 } { 3 } \right)$ 
B)  $\left( - \frac { 1 } { 2 } , 2 \right) , ( 1,5 )$ 
C)  $\left( \frac { 1 } { 3 } , \frac { 11 } { 3 } \right) , \left( - \frac { 3 } { 2 } , 0 \right)$ 
D)  $\left( \frac { 1 } { 6 } , \frac { 10 } { 3 } \right) , ( 1,5 )$ 
A)  $\left( \frac { 3 } { 2 } , 6 \right) , \left( - \frac { 1 } { 3 } , - \frac { 7 } { 3 } \right)$ 
B)  $\left( - \frac { 1 } { 2 } , 2 \right) , ( 1,5 )$ 
C)  $\left( \frac { 1 } { 3 } , \frac { 11 } { 3 } \right) , \left( - \frac { 3 } { 2 } , 0 \right)$ 
D)  $\left( \frac { 1 } { 6 } , \frac { 10 } { 3 } \right) , ( 1,5 )$

Question 9

Graph the system of inequalities.
-$\begin{array}{l}
y<x+1 \
2 x+3 y>6
\end{array}$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 10

Determine the real solutions to the system of nonlinear equations.
-$$\begin{array} { l } 
x ^ { 2 } - 3 y ^ { 2 } - 1 = 0 \
4 x ^ { 2 } + 3 y ^ { 2 } - 19 = 0
\end{array}$$

Accepted Answer

A)  $( 2,1 ) , ( 2 , - 1 ) , ( - 2,1 ) , ( - 2 , - 1 )$ 
B)  $( 2,1 ) , ( - 2 , - 1 )$ 
C)  $( - 1,2 ) , ( 1 , - 2 )$ 
D)  $( 1,2 ) , ( - 1,2 ) , ( 1 , - 2 ) , ( - 1 , - 2 )$ 
A)  $( 2,1 ) , ( 2 , - 1 ) , ( - 2,1 ) , ( - 2 , - 1 )$ 
B)  $( 2,1 ) , ( - 2 , - 1 )$ 
C)  $( - 1,2 ) , ( 1 , - 2 )$ 
D)  $( 1,2 ) , ( - 1,2 ) , ( 1 , - 2 ) , ( - 1 , - 2 )$

Question 11

Solve the system of linear equations using the elimination method.
-x = -5 - y - z 
x - y + 4z = 2 
2x + y =-4 - z

Accepted Answer

A) (1, -5, -1) 
B) (-1, 1, -5) 
C) (-1, -5, 1) 
D) (-5, -1, 1) 
A) (1, -5, -1) 
B) (-1, 1, -5) 
C) (-1, -5, 1) 
D) (-5, -1, 1)

Question 12

Graph the inequality.
-

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 13

Graph the inequality.
-

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 14

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
-$$\begin{array} { l } 
3 x - y = - 17 \
y = x ^ { 2 } + 7
\end{array}$$
(i) $( 5,32 )$; (ii) $( - 2,11 )$

Accepted Answer

A)  (i) Yes; (ii) No 
B)  (i) No; (ii) Yes 
C)  (i) No; (ii) No 
D)  (i) Yes; (ii) Yes 
A)  (i) Yes; (ii) No 
B)  (i) No; (ii) Yes 
C)  (i) No; (ii) No 
D)  (i) Yes; (ii) Yes

Question 15

Graph the system of inequalities.
-$\begin{array}{l}
x+2 y \geq 4 \
x+y \geq 2 \
y \geq 0 \
x \geq 0
\end{array}$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 16

Determine the real solutions to the system of nonlinear equations.
-$$\begin{array} { l } 
x ^ { 2 } + y ^ { 2 } = 50 \
( x - 2 ) ^ { 2 } + y ^ { 2 } = 50
\end{array}$$

Accepted Answer

A)  $( 1,7 ) , ( 1 , - 7 ) , ( - 1,7 ) , ( - 1 , - 7 )$ 
B)  $( 7,1 ) , ( 7 , - 1 )$ 
C)  $( 7,1 ) , ( 7 , - 1 ) , ( - 7,1 ) , ( - 7 , - 1 )$ 
D)  $( 1,7 ) , ( 1 , - 7 )$ 
A)  $( 1,7 ) , ( 1 , - 7 ) , ( - 1,7 ) , ( - 1 , - 7 )$ 
B)  $( 7,1 ) , ( 7 , - 1 )$ 
C)  $( 7,1 ) , ( 7 , - 1 ) , ( - 7,1 ) , ( - 7 , - 1 )$ 
D)  $( 1,7 ) , ( 1 , - 7 )$

Question 17

A system of nonlinear equations is given. Sketch the graph of the equation of the system and then determine the number of real solutions to the system. Do not solve the system.
-$$\begin{array} { l } 
x ^ { 2 } + y ^ { 2 } = 52 \
2 y + 3 x = 0
\end{array}$$

Accepted Answer

A)  Two
  
B)  Two
  
C)  Four
  
D)  Two
  
A)  Two
  
B)  Two
  
C)  Four
  
D)  Two

Question 18

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
-$$\begin{array} { l } 
x ^ { 2 } = y ^ { 2 } + 39 \
x - y = 3
\end{array}$$
(i) $( 8 , - 5 )$; (ii) $( - 8 , - 5 )$

Accepted Answer

A)  (i) No; (ii) $\mathrm { No }$ 
B)  (i) Yes; (ii) No 
C)  (i) Yes; (ii) Yes 
D)  (i) No; (ii) Yes 
A)  (i) No; (ii) $\mathrm { No }$ 
B)  (i) Yes; (ii) No 
C)  (i) Yes; (ii) Yes 
D)  (i) No; (ii) Yes

Question 19

Determine the real solutions to the system of nonlinear equations.
-$$\begin{array} { l } 
x ^ { 2 } - y ^ { 2 } = 27 \
x ^ { 2 } + y ^ { 2 } = 51
\end{array}$$

Accepted Answer

A)  $( 0,0 )$ 
B)  $( \sqrt { 39 } , 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , 2 \sqrt { 3 } )$ 
C)  $( \sqrt { 39 } , 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , 2 \sqrt { 3 } ) , ( \sqrt { 39 } , - 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , - 2 \sqrt { 3 } )$ 
D)  $( \sqrt { 39 } , 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , - 2 \sqrt { 3 } )$ 
A)  $( 0,0 )$ 
B)  $( \sqrt { 39 } , 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , 2 \sqrt { 3 } )$ 
C)  $( \sqrt { 39 } , 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , 2 \sqrt { 3 } ) , ( \sqrt { 39 } , - 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , - 2 \sqrt { 3 } )$ 
D)  $( \sqrt { 39 } , 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , - 2 \sqrt { 3 } )$

Question 20

Let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers.
-The sum of the squares of two numbers is 37. The difference of the two numbers is 7. Find the two numbers.

Accepted Answer

A) -6 and 1 
B) -6 and 1; -1 and 6 
C) -6 and -1; 1 and 6 
D) -1 and 6 
A) -6 and 1 
B) -6 and 1; -1 and 6 
C) -6 and -1; 1 and 6 
D) -1 and 6

Solve the problem. -The area of a garden is 9720 square feet, and the length of its diagonal is 153 feet. Find the dimensions of the garden.

Graph the system of inequalities. - y>-5 x\geq3 $Graph the system of inequalities. - \begin{array}{l} y>-5 \\ x \geq 3 \end{array}$

Write the partial fraction decomposition of the rational expression. - $\frac { 16 x ^ { 2 } - 105 x + 90 } { x ( x - 6 ) ( x - 5 ) }$

Graph the system of inequalities. - 2x+3y\geq6 x-y\geq3 y\leq2 $Graph the system of inequalities. - \begin{array}{r} 2 x+3 y \geq 6 \\ x-y \geq 3 \\ y \leq 2 \end{array}$

Use Gauss-Jordan elimination to solve the linear system and determine whether the system has a unique solution, no solution, or infinitely many solutions. If the system has infinitely many solutions, describe the solution as an ordered triple involving variable z. - x+y+z =7 x-y+2z =7 2x+3z =14

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - x+y+z=0 x-y+4z=0 2x+y+z=5

Solve the system of equations by substitution. -x + 7y = -2 3x + y = 34

Determine the real solutions to the system of nonlinear equations. - y=6-5x y=2x+3

Graph the system of inequalities. - y lt;x+1 2x+3y gt;6 $Graph the system of inequalities. - \begin{array}{l} y<x+1 \\ 2 x+3 y>6 \end{array}$

Determine the real solutions to the system of nonlinear equations. - -3-1=0 4+3-19=0

Solve the system of linear equations using the elimination method. -x = -5 - y - z x - y + 4z = 2 2x + y =-4 - z

Graph the inequality. -

Graph the inequality. -

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - 3x-y=-17 y=+7 (i) $( 5,32 )$ ; (ii) $( - 2,11 )$

Graph the system of inequalities. - x+2y\geq4 x+y\geq2 y\geq0 x\geq0 $Graph the system of inequalities. - \begin{array}{l} x+2 y \geq 4 \\ x+y \geq 2 \\ y \geq 0 \\ x \geq 0 \end{array}$

Determine the real solutions to the system of nonlinear equations. - +=50 (x-2+=50

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - =+39 x-y=3 (i) $( 8 , - 5 )$ ; (ii) $( - 8 , - 5 )$

Determine the real solutions to the system of nonlinear equations. - -=27 +=51

Let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. -The sum of the squares of two numbers is 37. The difference of the two numbers is 7. Find the two numbers.

Equations, Inequalities, and Applications

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Exam 7: Systems of Equations and Inequalities

Solve the problem. -The area of a garden is 9720 square feet, and the length of its diagonal is 153 feet. Find the dimensions of the garden.

Graph the system of inequalities. - y>-5 x\geq3

Write the partial fraction decomposition of the rational expression. - 16x2−105x+90x(x−6)(x−5)\frac { 16 x ^ { 2 } - 105 x + 90 } { x ( x - 6 ) ( x - 5 ) }x(x−6)(x−5)16x2−105x+90​

Graph the system of inequalities. - 2x+3y\geq6 x-y\geq3 y\leq2

Use Gauss-Jordan elimination to solve the linear system and determine whether the system has a unique solution, no solution, or infinitely many solutions. If the system has infinitely many solutions, describe the solution as an ordered triple involving variable z. - x+y+z =7 x-y+2z =7 2x+3z =14

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form. - x+y+z=0 x-y+4z=0 2x+y+z=5

Solve the system of equations by substitution. -x + 7y = -2 3x + y = 34

Determine the real solutions to the system of nonlinear equations. - y=6-5x y=2x+3

Graph the system of inequalities. - y lt;x+1 2x+3y gt;6

Determine the real solutions to the system of nonlinear equations. - -3-1=0 4+3-19=0

Solve the system of linear equations using the elimination method. -x = -5 - y - z x - y + 4z = 2 2x + y =-4 - z

Graph the inequality. -

Graph the inequality. -

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - 3x-y=-17 y=+7 (i) (5,32)( 5,32 )(5,32) ; (ii) (−2,11)( - 2,11 )(−2,11)

Graph the system of inequalities. - x+2y\geq4 x+y\geq2 y\geq0 x\geq0

Determine the real solutions to the system of nonlinear equations. - +=50 (x-2+=50

A system of nonlinear equations is given. Sketch the graph of the equation of the system and then determine the number of real solutions to the system. Do not solve the system. - +=52 2y+3x=0

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - =+39 x-y=3 (i) (8,−5)( 8 , - 5 )(8,−5) ; (ii) (−8,−5)( - 8 , - 5 )(−8,−5)

Determine the real solutions to the system of nonlinear equations. - -=27 +=51

Let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. -The sum of the squares of two numbers is 37. The difference of the two numbers is 7. Find the two numbers.

Equations, Inequalities, and Applications

The Rectangular Coordinate System, Lines, and Circles

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions and Equations

Conic Sections

Matrices

Sequences and Series; Counting and Probability

Math Exercises: Sets, Intervals, Absolute Value, and Properties

Filters

Graph the system of inequalities. - y>-5 x\geq3 $Graph the system of inequalities. - \begin{array}{l} y>-5 \\ x \geq 3 \end{array}$

Write the partial fraction decomposition of the rational expression. - $\frac { 16 x ^ { 2 } - 105 x + 90 } { x ( x - 6 ) ( x - 5 ) }$

Graph the system of inequalities. - 2x+3y\geq6 x-y\geq3 y\leq2 $Graph the system of inequalities. - \begin{array}{r} 2 x+3 y \geq 6 \\ x-y \geq 3 \\ y \leq 2 \end{array}$

Graph the system of inequalities. - y lt;x+1 2x+3y gt;6 $Graph the system of inequalities. - \begin{array}{l} y<x+1 \\ 2 x+3 y>6 \end{array}$

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - 3x-y=-17 y=+7 (i) $( 5,32 )$ ; (ii) $( - 2,11 )$

Graph the system of inequalities. - x+2y\geq4 x+y\geq2 y\geq0 x\geq0 $Graph the system of inequalities. - \begin{array}{l} x+2 y \geq 4 \\ x+y \geq 2 \\ y \geq 0 \\ x \geq 0 \end{array}$

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - =+39 x-y=3 (i) $( 8 , - 5 )$ ; (ii) $( - 8 , - 5 )$