Question 1

Solve the system of linear equations. $\left\{ \begin{array} { r } 
x + z = 2 \
y + 2 z = 7 \
- x + z = 4
\end{array} \right.$

Accepted Answer

$\left[ \begin{array} { c c c c } 
1 & 0 & 1 & 2 \
0 & 1 & 2 & 7 \
- 1 & 0 & 1 & 4
\end{array} \right]$ $\begin{array} { c } 
\Rightarrow \
R _ {3 } + R _ { 1 } \rightarrow R _ { 3 }
\end{array}$ $\left[ \begin{array} { l l l l } 
1 & 0 & 1 & 2 \
0 & 1 & 2 & 7 \
0 & 0 & 2 & 6
\end{array} \right]$ $\begin{array} { c } 
R _ { 1 } - \frac { 1 } { 2 } R _ {3 } \rightarrow R _ { 1 } \
\Rightarrow \
R _ { 2 } - R _ { 3 } \rightarrow R _ { 2 }
\end{array}$ $\left[ \begin{array} { c c c c } 
1 & 0 & 0 & - 1 \
0 & 1 & 0 & 1 \
0 & 0 & 2 & 6
\end{array} \right]$ $\begin{array} { c } 
\Rightarrow \
\frac { 1 } { 2 } R _ { 3 }
\end{array}$ $\left[ \begin{array} { c c c c } 
1 & 0 & 0 & - 1 \
0 & 1 & 0 & 1 \
0 & 0 & 1 & 3
\end{array} \right]$ , so the solution is $x = - 1$ , $y = 1$ , $z = 3$ .

Question 2

Graph the solution of the system of inequalities. $$\left\{ \begin{array} { l } 
x + 3 y > 5 \
2 x - y \leq 4
\end{array} \right.$$

Accepted Answer

The answer of Graph the solution of the system of...

Question 3

Graph the solution of the system of inequalities. $$\left\{ \begin{array} { l } 
y > 4 - x ^ { 2 } \
y \leq x
\end{array} \right.$$

Accepted Answer

The answer of Graph the solution of the system of...

Question 4

Given $A = \left[ \begin{array} { l l } 
6 & 5
\end{array} \right]$ , $B = \left[ \begin{array} { l l } 
- 5 & 4 \
6 & 7
\end{array} \right]$ , find $A B$ , or explain why the operation cannot be performed.

Accepted Answer

None

Question 5

Given $A = \left[ \begin{array} { c c } 
- 9 & 8 \
- 6 & - 5 \
10 & - 16
\end{array} \right]$ and $B = \left[ \begin{array} { l l } 
1 & 0 \
2 & 3 \
0 & 4
\end{array} \right]$ , solve $2 X - A = 3 B$ , for $X$ , or explain why there is no solution.

Accepted Answer

None

Question 6

Solve the system of equations $\left\{ \begin{aligned}
3 x + 2 y & = 8 \
x - 2 y & = 0
\end{aligned} \right.$ by converting to a matrix equation and using its inverse coefficient matrix $\left[ \begin{array} { c c } 
\frac { 1 } { 4 } & \frac { 1 } { 4 } \
\frac { 1 } { 8 } & - \frac { 3 } { 8 }
\end{array} \right]$ .

Accepted Answer

None

Question 7

Given $A = \left[ \begin{array} { l l } 
3 & 5 \
2 & 4
\end{array} \right]$ and $B = \left[ \begin{array} { c c } 
- 2 & 4 \
3 & 8
\end{array} \right]$ , solve $X - 2 A = B$ , for the matrix $X$ , or explain why there is no solution.

Accepted Answer

None

Question 8

Use the graphical method to find all the solutions of the system of equations. $\left\{ \begin{array} { l } 
y + 4 x = x ^ { 2 } \
y = - x ^ { 2 } + 8 x - 16
\end{array} \right.$

Accepted Answer

The soluti

Question 9

A money manager invests his client's savings into two accounts, one paying $2 \%$ and the other $10 \%$ simple interest per year. He puts three times as much in the lower yielding investment because it's less risky than the other one. His client's annual interest is $\$ 4625$ . How much did he invest at each rate?

Accepted Answer

Let @#LAT-DLM& be the amount invested at

Question 10

Solve the system of linear equations. $\left\{ \begin{array} { r } 
x + 15 y - z = 8 \
- x - 3 y + 2 z = 5 \
2 x + 6 y - 4 z = - 10
\end{array} \right.$

Accepted Answer

The answer of Solve the system of linear equations. \(\left\{...

Question 11

Solve the system of equations $\left\{ \begin{array} { r } 
x + 2 y + z + w = 0 \
x + 2 y - z - w = 0 \
x + 4 y + z + 2 w = - 1 \
x + 3 y + z + w = 2
\end{array} \right.$ by converting to a matrix equation and using its inverse coefficient matrix $\left[ \begin{array} { c c c c } 
\frac { 5 } { 2 } & \frac { 1 } { 2 } & 0 & - 2 \
- 1 & 0 & 0 & 1 \
- \frac { 1 } { 2 } & - \frac { 1 } { 2 } & - 1 & 2 \
1 & 0 & 1 & - 2
\end{array} \right]$ .

Accepted Answer

None

Question 12

Find all solutions for the system. $\left\{ \begin{array} { l } 
\frac { 4 } { x } - \frac { 5 } { y } = 2 \
- \frac { 1 } { x } + \frac { 3 } { y } = 1
\end{array} \right.$

Accepted Answer

The answer of Find all solutions for the system. \(\left\{...

Question 13

Find the determinant of the matrix. Determine whether the matrix has an inverse or not, but do not calculate the inverse. $\left| \begin{array} { c c c } 
2 & 1 & 6 \
- 2 & 5 & - 3 \
8 & 1 & 5
\end{array} \right|$

Accepted Answer

. The matr

Question 14

Use back-substitution to solve the triangular system. $$\left\{ \begin{array} { r } 
x + 2 y + z = 0 \
y - 3 z = 1 \
4 z = - 4
\end{array} \right.$$

Accepted Answer

The answer of Use back-substitution to solve the triangular system....

Question 15

Find the complete solution of the linear system or show that it is inconsistent. $$\left\{ \begin{array} { r } 
x + 2 y + w = 3 \
3 x + 2 y - z - w = 0 \
y + 2 w = 1 \
y + z + w = 3
\end{array} \right.$$

Accepted Answer

The answer of Find the complete solution of the linear...

Question 16

State whether the equation $3 x - 5 y + z = 42$ is linear.

Accepted Answer

The answer of State whether the equation \(3 x -...

Question 17

Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\{ \begin{array} { l } 
3 x - 4 y = 12 \
- x + \frac { 4 } { 3 } y = 4
\end{array} \right.$$

Accepted Answer

The answer of Graph each linear system, either by hand...

Question 18

Given $A = \left[ \begin{array} { c c c } 
1 & - 3 & 2 \
4 & - 1 & 3 \
0 & 5 & 2
\end{array} \right]$ and $B = \left[ \begin{array} { c c c } 
0 & 3 & - 2 \
2 & 1 & 0 \
- 3 & - 4 & 2
\end{array} \right]$ , find $A B$ , or explain why the operation cannot be performed.

Accepted Answer

None

Question 19

Given $A = \left[ \begin{array} { c c c } 
2 & 5 & - 3 \
3 & - 2 & 0 \
4 & 2 & 2
\end{array} \right]$ and $B = \left[ \begin{array} { c c } 
- 1 & 5 \
0 & 25
\end{array} \right]$ , find $A - B$ , or explain why the operation cannot be performed.

Accepted Answer

The operation cannot

Question 20

Graph the inequality $y > - x$ .

Accepted Answer

The answer of Graph the inequality $y > - x$...

Solve the system of linear equations. $\left\{ \begin{array} { r } x + z = 2 \\y + 2 z = 7 \\- x + z = 4\end{array} \right.$

Graph the solution of the system of inequalities. $\left\{ \begin{array} { l } x + 3 y > 5 \\2 x - y \leq 4\end{array} \right.$

Graph the solution of the system of inequalities. $\left\{ \begin{array} { l } y > 4 - x ^ { 2 } \\y \leq x\end{array} \right.$

Given $A = \left[ \begin{array} { l l } 6 & 5\end{array} \right]$ , $B = \left[ \begin{array} { l l } - 5 & 4 \\6 & 7\end{array} \right]$ , find $A B$ , or explain why the operation cannot be performed.

Given $A = \left[ \begin{array} { c c } - 9 & 8 \\- 6 & - 5 \\10 & - 16\end{array} \right]$ and $B = \left[ \begin{array} { l l } 1 & 0 \\2 & 3 \\0 & 4\end{array} \right]$ , solve $2 X - A = 3 B$ , for $X$ , or explain why there is no solution.

Given $A = \left[ \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right]$ and $B = \left[ \begin{array} { c c } - 2 & 4 \\3 & 8\end{array} \right]$ , solve $X - 2 A = B$ , for the matrix $X$ , or explain why there is no solution.

Use the graphical method to find all the solutions of the system of equations. $\left\{ \begin{array} { l } y + 4 x = x ^ { 2 } \\y = - x ^ { 2 } + 8 x - 16\end{array} \right.$

Solve the system of linear equations. $\left\{ \begin{array} { r } x + 15 y - z = 8 \\- x - 3 y + 2 z = 5 \\2 x + 6 y - 4 z = - 10\end{array} \right.$

Find all solutions for the system. $\left\{ \begin{array} { l } \frac { 4 } { x } - \frac { 5 } { y } = 2 \\- \frac { 1 } { x } + \frac { 3 } { y } = 1\end{array} \right.$

Find the determinant of the matrix. Determine whether the matrix has an inverse or not, but do not calculate the inverse. $\left| \begin{array} { c c c } 2 & 1 & 6 \\- 2 & 5 & - 3 \\8 & 1 & 5\end{array} \right|$

Use back-substitution to solve the triangular system. $\left\{ \begin{array} { r } x + 2 y + z = 0 \\y - 3 z = 1 \\4 z = - 4\end{array} \right.$

Find the complete solution of the linear system or show that it is inconsistent. $\left\{ \begin{array} { r } x + 2 y + w = 3 \\3 x + 2 y - z - w = 0 \\y + 2 w = 1 \\y + z + w = 3\end{array} \right.$

State whether the equation $3 x - 5 y + z = 42$ is linear.

Given $A = \left[ \begin{array} { c c c } 1 & - 3 & 2 \\4 & - 1 & 3 \\0 & 5 & 2\end{array} \right]$ and $B = \left[ \begin{array} { c c c } 0 & 3 & - 2 \\2 & 1 & 0 \\- 3 & - 4 & 2\end{array} \right]$ , find $A B$ , or explain why the operation cannot be performed.

Given $A = \left[ \begin{array} { c c c } 2 & 5 & - 3 \\3 & - 2 & 0 \\4 & 2 & 2\end{array} \right]$ and $B = \left[ \begin{array} { c c } - 1 & 5 \\0 & 25\end{array} \right]$ , find $A - B$ , or explain why the operation cannot be performed.

Graph the inequality $y > - x$ .

Fundamentals

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Unit Circle Approach

Trigonometric Functions: Right Triangle Approach

Analytic Trigonometry

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Conic Sections

Sequences and Series

Limits: a Preview of Calculus

Filters

Exam 10: Systems of Equations and Inequalities

Solve the system of linear equations. {x+z=2y+2z=7−x+z=4\left\{ \begin{array} { r } x + z = 2 \\y + 2 z = 7 \\- x + z = 4\end{array} \right.⎩⎨⎧​x+z=2y+2z=7−x+z=4​

Graph the solution of the system of inequalities. {x+3y>52x−y≤4\left\{ \begin{array} { l } x + 3 y > 5 \\2 x - y \leq 4\end{array} \right.{x+3y>52x−y≤4​

Graph the solution of the system of inequalities. {y>4−x2y≤x\left\{ \begin{array} { l } y > 4 - x ^ { 2 } \\y \leq x\end{array} \right.{y>4−x2y≤x​

Given A=[65]A = \left[ \begin{array} { l l } 6 & 5\end{array} \right]A=[6​5​] , B=[−5467]B = \left[ \begin{array} { l l } - 5 & 4 \\6 & 7\end{array} \right]B=[−56​47​] , find ABA BAB , or explain why the operation cannot be performed.

Given A=[3524]A = \left[ \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right]A=[32​54​] and B=[−2438]B = \left[ \begin{array} { c c } - 2 & 4 \\3 & 8\end{array} \right]B=[−23​48​] , solve X−2A=BX - 2 A = BX−2A=B , for the matrix XXX , or explain why there is no solution.

Use the graphical method to find all the solutions of the system of equations. {y+4x=x2y=−x2+8x−16\left\{ \begin{array} { l } y + 4 x = x ^ { 2 } \\y = - x ^ { 2 } + 8 x - 16\end{array} \right.{y+4x=x2y=−x2+8x−16​

Solve the system of linear equations. {x+15y−z=8−x−3y+2z=52x+6y−4z=−10\left\{ \begin{array} { r } x + 15 y - z = 8 \\- x - 3 y + 2 z = 5 \\2 x + 6 y - 4 z = - 10\end{array} \right.⎩⎨⎧​x+15y−z=8−x−3y+2z=52x+6y−4z=−10​

Find all solutions for the system. {4x−5y=2−1x+3y=1\left\{ \begin{array} { l } \frac { 4 } { x } - \frac { 5 } { y } = 2 \\- \frac { 1 } { x } + \frac { 3 } { y } = 1\end{array} \right.{x4​−y5​=2−x1​+y3​=1​

Find the determinant of the matrix. Determine whether the matrix has an inverse or not, but do not calculate the inverse. ∣216−25−3815∣\left| \begin{array} { c c c } 2 & 1 & 6 \\- 2 & 5 & - 3 \\8 & 1 & 5\end{array} \right|​2−28​151​6−35​​

Use back-substitution to solve the triangular system. {x+2y+z=0y−3z=14z=−4\left\{ \begin{array} { r } x + 2 y + z = 0 \\y - 3 z = 1 \\4 z = - 4\end{array} \right.⎩⎨⎧​x+2y+z=0y−3z=14z=−4​

Find the complete solution of the linear system or show that it is inconsistent. {x+2y+w=33x+2y−z−w=0y+2w=1y+z+w=3\left\{ \begin{array} { r } x + 2 y + w = 3 \\3 x + 2 y - z - w = 0 \\y + 2 w = 1 \\y + z + w = 3\end{array} \right.⎩⎨⎧​x+2y+w=33x+2y−z−w=0y+2w=1y+z+w=3​

State whether the equation 3x−5y+z=423 x - 5 y + z = 423x−5y+z=42 is linear.

Graph the inequality y>−xy > - xy>−x .

Fundamentals

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Unit Circle Approach

Trigonometric Functions: Right Triangle Approach

Analytic Trigonometry

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Conic Sections

Sequences and Series

Limits: a Preview of Calculus

Filters

Solve the system of linear equations. $\left\{ \begin{array} { r } x + z = 2 \\y + 2 z = 7 \\- x + z = 4\end{array} \right.$

Graph the solution of the system of inequalities. $\left\{ \begin{array} { l } x + 3 y > 5 \\2 x - y \leq 4\end{array} \right.$

Graph the solution of the system of inequalities. $\left\{ \begin{array} { l } y > 4 - x ^ { 2 } \\y \leq x\end{array} \right.$

Given $A = \left[ \begin{array} { l l } 6 & 5\end{array} \right]$ , $B = \left[ \begin{array} { l l } - 5 & 4 \\6 & 7\end{array} \right]$ , find $A B$ , or explain why the operation cannot be performed.

Given $A = \left[ \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right]$ and $B = \left[ \begin{array} { c c } - 2 & 4 \\3 & 8\end{array} \right]$ , solve $X - 2 A = B$ , for the matrix $X$ , or explain why there is no solution.

Use the graphical method to find all the solutions of the system of equations. $\left\{ \begin{array} { l } y + 4 x = x ^ { 2 } \\y = - x ^ { 2 } + 8 x - 16\end{array} \right.$

Solve the system of linear equations. $\left\{ \begin{array} { r } x + 15 y - z = 8 \\- x - 3 y + 2 z = 5 \\2 x + 6 y - 4 z = - 10\end{array} \right.$

Find all solutions for the system. $\left\{ \begin{array} { l } \frac { 4 } { x } - \frac { 5 } { y } = 2 \\- \frac { 1 } { x } + \frac { 3 } { y } = 1\end{array} \right.$

Find the determinant of the matrix. Determine whether the matrix has an inverse or not, but do not calculate the inverse. $\left| \begin{array} { c c c } 2 & 1 & 6 \\- 2 & 5 & - 3 \\8 & 1 & 5\end{array} \right|$

Use back-substitution to solve the triangular system. $\left\{ \begin{array} { r } x + 2 y + z = 0 \\y - 3 z = 1 \\4 z = - 4\end{array} \right.$

Find the complete solution of the linear system or show that it is inconsistent. $\left\{ \begin{array} { r } x + 2 y + w = 3 \\3 x + 2 y - z - w = 0 \\y + 2 w = 1 \\y + z + w = 3\end{array} \right.$

State whether the equation $3 x - 5 y + z = 42$ is linear.

Graph the inequality $y > - x$ .