123 - FML #1 - 18.1W - e.g. 20

Solving Linear Systems

6 Activities

Wize University Linear Algebra Textbook > Systems of Linear Equations (SLEs) (Linear Systems)

Augmented Matrix

5 Activities

Mark Yourself Question

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Consider the following system of linear equations: 
		{x−2y+4z=22x+y−2z=−13x−y+2z=12x+6y−12z=−6\bcb{

\begin{cases}
					x - 2y + 4z = 2& \\
					2x + y - 2z = -1 &\\
					3x - y + 2z = 1 &\\
					2x + 6y - 12z = -6& \end{cases}}⎩⎨⎧​x−2y+4z=22x+y−2z=−13x−y+2z=12x+6y−12z=−6​​				
a) Construct the augmented matrix A\bcb{A}A corresponding to this system.
b) Solve this system using row operations on A\bcb{A}A. How many solutions are there, if any?
			

I have answered this question

Find size of soln set to nonhom linear system

Find all solutions to the system of linear equations 5x+y+z=0x+2z=−12x+y−5z=1\begin{array}{rcl}
5x+y+z=0\\
x+2z=-1\\
2x+y-5z=1
\end{array}5x+y+z=0x+2z=−12x+y−5z=1​.

Given the augmented matrix  [1201010200k2−9k−3]\left[\begin{array}{rrr|r}
1 & 2 &0 &1  \\

0 & 1  &0&2\\
0&0&k^2-9&k-3
 \end{array}\right]​100​210​00k2−9​12k−3​​, answer the following questions.

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 133 - FML 1 - 18.1W e.g. 44.4

Interpret the following matrix as the RREF of a matrix of coefficients augmented by the vector of constants and determine whether the systems  has a unique solution.
[100001000010] \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} ​100​010​001​000​​

 Find the all values of p and q such that the system has at least one solution.
[12401300p∣15q]\left[\begin{array}{l}
1&2&4\\
0&1&3\\
0&0&p\\
\end{array}\right.\left|\begin{array}{l}1
\\5\\q\end{array}\right]​100​210​43p​​15q​​

 Find the value of p such that there is a unique solution to this system of equations:
L:{         2x1+x3=4x1+3x2+px3=−10     x1−x2+3x3=5L:\begin{cases}
\ \ \ \ \ \ \ \ \ 2x_1+x_3=4\\
x_1+3x_2+px_3=-10\\
 \ \ \ \ \ x_1-x_2+3x_3=5
\end{cases}L:⎩⎨⎧​         2x1​+x3​=4x1​+3x2​+px3​=−10     x1​−x2​+3x3​=5​

a) For what values of c does the system have one solution?
{2𝑥+𝒄𝑦=0𝒄𝑥+𝑦=3\begin{cases}
2𝑥 + 𝒄𝑦 = 0\\
𝒄𝑥 + 𝑦 = 3
\end{cases}{2x+cy=0cx+y=3​
b) For what values of c does it have infinitely many solutions?

For what value of p does the following system of equations have no solution? What is the geometric interpretation of these equations?
x+3y=1x+3y=1x+3y=1
px−6y=−4px-6y=-4px−6y=−4

Practice Question: Solving a System of Linear Equations (Infinitely Many Solutions)

Practice Question: Solving a SLE
Solve the following system of linear equations: x+2y=4y+z=3z+w=1\begin{array}{rcl}
x+2y&=&4\\
y+z&=&3\\
z+w&=&1
\end{array}x+2yy+zz+w​===​431​.

The row echelon form of the augmented matrix for a linear system is:
[A∣b][10111012400000100000∣1258].[A|b]\left[\begin{array}{l}
1&0&1&1&1\\
0&1&2&4&0\\
0&0&0&0&1\\
0&0&0&0&0
\end{array}\right.\left|\begin{array}{l}1
\\2\\5\\8\end{array}\right]_.[A∣b]​1000​0100​1200​1400​1010​​1258​​.​

Which of the following options is the general solution of the following system of linear equations?
x−y−z=2x+y+z=0\begin{array}{l}x-y-z=2\\x+y+z=0\end{array}x−y−z=2x+y+z=0​

Find the basic solutions of the following homogeneous system of linear equations:
x1−2x2+3x3+x4+2x5=0x1−x2+3x3−x4+3x5=0−x1+2x2−4x3+x4+x5=0\begin{array}{l}x_1 - 2x_2 + 3x_3 + x_4 + 2x_5 = 0\\x_1 - x_2 + 3x_3 - x_4 + 3x_5 = 0\\-x_1 + 2x_2 - 4x_3 + x_4 + x_5 = 0\end{array}x1​−2x2​+3x3​+x4​+2x5​=0x1​−x2​+3x3​−x4​+3x5​=0−x1​+2x2​−4x3​+x4​+x5​=0​

What can you say about the following systems of linear equations?
x−2y−3z=42x+2y−6z=−1\begin{array}{l}x-2y-3z=4\\2x+2y-6z=-1\end{array}x−2y−3z=42x+2y−6z=−1​

What can you say about the following systems of linear equations?
x−y+3z=52x−2y+6z=−10\begin{array}{l}x-y+3z=5\\2x-2y+6z=-10\end{array}x−y+3z=52x−2y+6z=−10​

Write the general solution to the system of linear equations
x−y−3z=−42x−3y+z=0\begin{array}{l} x - y - 3z = -4\\ 2x - 3y + z = 0 \end{array}x−y−3z=−42x−3y+z=0​

Wize Concept Clarifier--Gaussian Jordan Elimination

Wize Concept Clarifier
Solve the system x+y−z=24y−4z=12x+3y−3z=−1\begin{array}{}
x+y-z=2\\
4y-4z=1\\
2x+3y-3z=-1
\end{array}x+y−z=24y−4z=12x+3y−3z=−1​.

The row echelon form of the augmented matrix for a linear system is:
[A∣b][10111012400000100000∣1258].[A|b]\left[\begin{array}{l}
1&0&1&1&1\\
0&1&2&4&0\\
0&0&0&0&1\\
0&0&0&0&0
\end{array}\right.\left|\begin{array}{l}1
\\2\\5\\8\end{array}\right]_.[A∣b]​1000​0100​1200​1400​1010​​1258​​.​

Linear Systems - Mixing

You are the cook on a celebrity yacht, and you have run out of flour, sugar, and baking powder -
except for that contained in a large quantity of 3 different pancake mixes:
Mix A: 8 parts flour to 8 parts sugar to 1 part baking powder

Basics of Systems of Linear Equations

Practice: Solutions to Linear Systems
Determine which vectors are solutions to the following SLE [select all that apply]:
x1 − 5x2 +x3=2−x1 +5x2 −3x3= −6\begin{alignedat}{8}
x_1\ &-&\ 5x_2\ &+& x_3 &=& 2\\
-x_1\ &+& 5x_2\ &-& 3x_3 &=&\ -6
\end{alignedat}x1​ −x1​ ​−+​ 5x2​ 5x2​ ​+−​x3​3x3​​==​2 −6​

Solve linear systems

Solve the following systems of linear equations:
a) x+2y=−3−x+3y=1\begin{array}{r}x+2y=-3\\-x+3y=1\end{array}x+2y=−3−x+3y=1​
b) x+y−2z=1−4x+3y+2z=1−3x+11x−6z=0\begin{array}{r}
x+y-2z=1\\
-4x+3y+2z=1\\
-3x+11x-6z=0
\end{array}x+y−2z=1−4x+3y+2z=1−3x+11x−6z=0​

123 - FML #1 - 18.1W - e.g. 21.1

Solve the following system of linear equations:
 {2x+y+z+3w=13x−2z+2w=−15x+y−z+5w=07x+2y+8w=1\bcb{ \begin{cases}
										2x + y + z + 3w = 1 \\
										3x - 2z + 2w = -1 \\
										5x + y - z + 5w = 0 \\
										7x + 2y + 8w = 1
										\end{cases} }⎩⎨⎧​2x+y+z+3w=13x−2z+2w=−15x+y−z+5w=07x+2y+8w=1​ 

Solve the following linear system of equations by Gaussian Elimination.
2x−y+5z=14x+y+z=4−x−3y+3z=−2\begin{array}{c}
2x - y + 5z = 14\\
x + y + z = 4\\
-x - 3y + 3z = - 2
\end{array}2x−y+5z=14x+y+z=4−x−3y+3z=−2​

Solve the following linear system of equations.
3x+2y−2z=185x+2y−z=27x+y+z=9\begin{array}{c}
3x + 2y - 2z = 18\\
5x + 2y - z =27\\
x + y + z = 9
\end{array}3x+2y−2z=185x+2y−z=27x+y+z=9​

Find soln to nonhom linear system

Find all solutions to the following system of linear equations[10210131]\left[\begin{array}{rrr|r}
1&0&2&1\\
0&1&3&1
\end{array}\right][10​01​23​11​].

Balancing Chemical Equations

The chemical equation for the combustion of octane (the primary component of gasoline) is:
C8H18+O2=H2O+CO2C_8H_{18}+O_2 = H_2O+CO_2C8​H18​+O2​=H2​O+CO2​
Balance this equation using Gauss-Jordan elimination

Find all possible solutions to the following linear system of equations.
x+2y−z=9x−y+2z=3\begin{array}{c}
x + 2y - z = 9\\
x - y + 2z = 3
\end{array}x+2y−z=9x−y+2z=3​

Solving Linear Systems

Practice: Solving a Linear System
Solve the following system of linear equations. What type of solution is it?
3x1+6x2=12x1+4x2=2x2+x3=3\begin{aligned}
3x_1 + 6x_2&= 1\\
2x_1 + 4x_2 &=2\\
x_2 + x_3 &= 3  
\end{aligned}3x1​+6x2​2x1​+4x2​x2​+x3​​=1=2=3​

Solving Linear Systems

Practice: Solving a Homogeneous Linear System
Solve the following homogeneous system of linear equations. What type of solution is it?
2x1−x2+2x4=02x2+4x3−x4=0\begin{aligned}
2x_1-x_2+2x_4 &= 0\\
2x_2 + 4x_3 − x_4 &= 0
\end{aligned}2x1​−x2​+2x4​2x2​+4x3​−x4​​=0=0​

Solving Linear Systems

Practice: Geometric Interpretation of a Linear System
Consider the following augmented matrix in RREF:
[1−100−540010−2200014−3]\left[\begin{array}{rrrrr|r}
1&-1&0&0&-5&4\\[0.5em]
0&0&1&0&-2&2\\[0.5em]
0&0&0&1&4&-3
\end{array}\right]​100​−100​010​001​−5−24​42−3​​

 Find I0 in the following circuit.

Solving Linear Systems

Example: Solving a Linear System (Unique Solution)
Solve the system of linear equations:
2x+2y−2z=2x+2z−7=03y=−7−2z\begin{array}{ccccccl}
2x&+&2y&-&2z&=&2\\
x&+&2z&-&7&=&0\\
&&&&3y &=&-7-2z
\end{array}2xx​++​2y2z​−−​2z73y​===​20−7−2z​

Solving Linear Systems

Example: Solving a Linear System (Infinitely Many Solutions)
Solve the following system of linear equations:
x+2y=4y+z=3z+w=1\begin{array}{c}
x+2y&=&4\\
y+z&=&3\\
z+w&=&1
\end{array}x+2yy+zz+w​===​431​

19.4F_Final_Builder_Ch_2.13_Algebra_SLE_$\tkco{eg22}$_$\key{Final}$_Builder_$\tkcth{2.13.}\tkcf{22}$_

Given the system of equations:
{2x1−4x2=−1−4x1+8x2=2
            \left\{
            \begin{array}{ll}
              2x_1 - 4x_2 & = \quad -1 \\
              -4x_1 + 8x_2 & = \quad 2
            \end{array}
            \right.
        {2x1​−4x2​−4x1​+8x2​​=−1=2​
Show that x⃗g=x⃗p+tx⃗h\vx_g = \vx_p + t \vx_hxg​=xp​+txh​, where x⃗p=[−120]\vx_p = \colvec{-\frac12}{0}xp​=[−21​0​] and x⃗h=[21]\vx_h = \colvec{2}{1}xh​=[21​] is a solution to the system when t=1,2,3t = 1, 2, 3t=1,2,3, i.e. show that the vectors:

Find (if possible) conditions on aaa and bbb so that the following system has
(a) no solution
(b) one solution

Which of the following options is the general solution of the following system of linear equations?
x−2y+z=3x−y−z=0\begin{array}{l}x-2y+z=3\\x-y-z=0\end{array}x−2y+z=3x−y−z=0​

What can you say about the following systems of linear equations?
x+y−2z=22x+2y−6z=3\begin{array}{l}x+y-2z=2\\2x+2y-6z=3\end{array}x+y−2z=22x+2y−6z=3​

What can you say about the following systems of linear equations?
x+y−2z=−32x+2y−4z=−6\begin{array}{l}x+y-2z=-3\\2x+2y-4z=-6\end{array}x+y−2z=−32x+2y−4z=−6​

Interpretation of Linear Systems

Which of the following describes the intersection between the following 3 hyperplanes in R4\reals^4R4?
2x1+3x3+x4=1x2−x4=0x2=−3\begin{array}{rcl}
2x_1+3x_3+x_4&=&1\\
x_2-x_4&=&0\\
x_2&=&-3
\end{array}2x1​+3x3​+x4​x2​−x4​x2​​===​10−3​

Matrix Multiplication

e.g. Find a 3 × 3 matrix M such that 
 ⁣M[   0−2   0]=[−1   1−1] ⁣, ⁣M[001]=[   0−2   3] ⁣,and M[1300]=[−2   2−1]\!M\begin{bmatrix}
\ \ \ 0\\-2\\\ \ \ 0
\end{bmatrix}=
\begin{bmatrix}
-1\\\ \ \ 1\\-1
\end{bmatrix}\!, \hspace{0.7cm}

\!M\begin{bmatrix}
0\\0\\1
\end{bmatrix}=
\begin{bmatrix}
\ \ \ 0\\-2\\\ \ \ 3
\end{bmatrix}\!, \hspace{0.7cm} \text{\small{and}}\ 

M\begin{bmatrix}
\frac{1}{3}\\0\\0
\end{bmatrix}=
\begin{bmatrix}
-2\\\ \ \ 2\\-1
\end{bmatrix}M​   0−2   0​​=​−1   1−1​​,M​001​​=​   0−2   3​​,and M​31​00​​=​−2   2−1​​

System of Linear Equations

Find (if possible) conditions on aaa and bbb so that the following system has
(a) no solution
(b) one solution

Homogeneous System of Equations

Find basic solutions of a homogeneous system of linear equations with reduced row echelon form of:
[1203000150]\left[\begin{array}{cccc|c}1 & 2 & 0 & 3 & 0 \\0 & 0 & 1 & 5 & 0\end{array}\right][10​20​01​35​00​]

Practice Question: Solving a System of Linear Equations (Unique Solution)

Practice Question: Solving a SLE--Unique Solution
Solve the following systems of linear equations.
   𝑥−𝑧=5𝑥−𝑧=5x−z=5

Find size of soln set to nonhom linear system

Find all solutions to the system of linear equations 5x+y+z=0x+2z=−12x+y−5z=1\begin{array}{rcl}
5x+y+z=0\\
x+2z=-1\\
2x+y-5z=1
\end{array}5x+y+z=0x+2z=−12x+y−5z=1​.

What can you say about the following systems of linear equations?
x+y−2z=22x+2y−6z=3\begin{array}{l}x+y-2z=2\\2x+2y-6z=3\end{array}x+y−2z=22x+2y−6z=3​

Write the general solution (all solutions) to the system of linear equations
x−y−3z=−42x−3y+z=0\begin{array}{l}x-y-3z=-4\\2x-3y+z=0\end{array}x−y−3z=−42x−3y+z=0​

Find (if possible) conditions on aaa and bbb so that the following system has
(a) no solution
(b) one solution

What can you say about the following systems of linear equations?
x+y−2z=−32x+2y−4z=−6\begin{array}{l}x+y-2z=-3\\2x+2y-4z=-6\end{array}x+y−2z=−32x+2y−4z=−6​

Which of the following options is the general solution of the following system of linear equations?
x−2y+z=3x−y−z=0\begin{array}{l}x-2y+z=3\\x-y-z=0\end{array}x−2y+z=3x−y−z=0​

Find all solutions to the system of linear equations 5x+y+z=0x+2z=−12x+y−5z=1\begin{array}{c}
5x+y+z=0\\
x+2z=-1\\
2x+y-5z=1
\end{array}5x+y+z=0x+2z=−12x+y−5z=1​.

Practice Question: Solving a System of Linear Equations (Infinitely Many Solutions)

Practice Question: Solving a SLE
Solve the following system of linear equations: x+2y=4y+z=3z+w=1\begin{array}{rcl}
x+2y&=&4\\
y+z&=&3\\
z+w&=&1
\end{array}x+2yy+zz+w​===​431​.

Solve the homogeneous linear system

When the right hand side of every equation in a linear system is 0, we say that the system is homogeneous 
Find the general solution of the following homogeneous linear system:
{3x+2y=0−6x+4y=0\left\{\begin{array}{rcl}
3x+2y&=&0\\
-6x+4y&=&0
\end{array}
\right.{3x+2y−6x+4y​==​00​

Solving a system of linear equations

Solve the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr}
-2x_1+3x_2+13x_3&=&5\\
-x_1+2x_2+5x_3&=&1\\
4x_1-14x_2-2x_3&=&14\\
\end{array}−2x1​+3x2​+13x3​−x1​+2x2​+5x3​4x1​−14x2​−2x3​​===​5114​

Solving a system of linear equations

Solve the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr}
-2x_1+3x_2+13x_3&=&5\\
-x_1+2x_2+5x_3&=&1\\
4x_1-14x_2-2x_3&=&14\\
\end{array}−2x1​+3x2​+13x3​−x1​+2x2​+5x3​4x1​−14x2​−2x3​​===​5114​

Solving a system of linear equations

Solve the following system of linear equations:
−2x1−4x2−6x3−19x4=−153x1+10x2−30x3+9x4=24x1+3x2−6x3+4x4=87x1+17x2+45x4=55\begin{array}{rcr}
-2x_1-4x_2-6x_3-19x_4&=&-15\\
3x_1+10x_2-30x_3+9x_4&=&24\\
x_1+3x_2-6x_3+4x_4&=&8\\
7x_1+17x_2+45x_4&=&55
\end{array}−2x1​−4x2​−6x3​−19x4​3x1​+10x2​−30x3​+9x4​x1​+3x2​−6x3​+4x4​7x1​+17x2​+45x4​​====​−1524855​
where we wish to solve for the quadruple (x1,x2,x3,x4)(x_1,x_2,x_3,x_4)(x1​,x2​,x3​,x4​) of real numbers that satisfy this system

Practice: Solutions to a System of Equations

Practice: Solutions to a System of Equations
Given the system of linear equations x+y=0x−cy=1\begin{array}{c}
x+y=0\\
x-cy=1
\end{array}x+y=0x−cy=1​, find the values of ccc such that,

Solving a system of linear equations

Solve the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr}
-2x_1+3x_2+13x_3&=&5\\
-x_1+2x_2+5x_3&=&1\\
4x_1-14x_2-2x_3&=&14\\
\end{array}−2x1​+3x2​+13x3​−x1​+2x2​+5x3​4x1​−14x2​−2x3​​===​5114​
where we wish to solve for the triple (x1,x2,x3)(x_1,x_2,x_3)(x1​,x2​,x3​) of real numbers that satisfy this system

Practice: Solving a System of Equations

Practice: Solving a System of Equations
Solve the system of equations 2x+2y−2z=2x+2z−7=03y=−7−2z\begin{array}{rcl}
2x+2y-2z&=&2\\
x+2z-7&=&0\\
3y&=&-7-2z
\end{array}2x+2y−2zx+2z−73y​===​20−7−2z​

Linear algebra

For which values of aaaand bbbdoes the system x+3y=2b3x−3ay=a\begin{array}{l}x+3y = 2b\\3x-3ay = a\end{array}x+3y=2b3x−3ay=a​have infinitely many solutions?

For which values of a and b does the system x+3y=2b3x−3ay=a\begin{array}{l}x+3y = 2b\\3x-3ay = a\end{array}x+3y=2b3x−3ay=a​have infinitely many solutions?

For which values of a and b does the system 2x−4y=a3x−ay=3b\begin{array}{l}2x-4y = a\\3x-ay = 3b\end{array}2x−4y=a3x−ay=3b​have no solutions?

Write the general solution to the system of linear equations
x−y−3z=−42x−3y+z=0\begin{array}{l} x - y - 3z = -4\\ 2x - 3y + z = 0 \end{array}x−y−3z=−42x−3y+z=0​

System of Linear Equations

Write the general solution to the system of linear equations
x−y−3z=−42x−3y+z=0\begin{array}{l} x - y - 3z = -4\\ 2x - 3y + z = 0 \end{array}x−y−3z=−42x−3y+z=0​

Solve the following linear system of equations.
2x+3y−z=7x−2y+5z=27y−11z=6\begin{array}{c}
2x + 3y - z = 7\\
x - 2y + 5z = 2\\
7y - 11z = 6
\end{array}2x+3y−z=7x−2y+5z=27y−11z=6​

Solving Linear Systems

Solve the following homogeneous system of linear equations. What type of solution is it?
2x1−x2+2x4=02x2+4x3−x4=0\begin{aligned}
2x_1-x_2+2x_4 &= 0\\
2x_2 + 4x_3 − x_4 &= 0
\end{aligned}2x1​−x2​+2x4​2x2​+4x3​−x4​​=0=0​

Solving Linear Systems

Solve the following system of linear equations. What type of solution is it?
3x1+6x2=12x1+4x2=2x2+x3=3\begin{aligned}
3x_1 + 6x_2&= 1\\
2x_1 + 4x_2 &=2\\
x_2 + x_3 &= 3  
\end{aligned}3x1​+6x2​2x1​+4x2​x2​+x3​​=1=2=3​

Interpretation of Linear Systems

Which of the following describes the intersection between the following 3 hyperplanes in R4\reals^4R4?
2x1+3x3+x4=1x2−x4=0x2=−3\begin{array}{rcl}
2x_1+3x_3+x_4&=&1\\
x_2-x_4&=&0\\
x_2&=&-3
\end{array}2x1​+3x3​+x4​x2​−x4​x2​​===​10−3​

Writing the augmented matrix of a system of linear equations

Write the augmented matrix for the following system:
−x1+2x3=5−x25x1+x2=9+x310x3+2x2=1−x1\begin{array}{rcr}
-x_1+2x_3&=&5-x_2\\
5x_1+x_2&=&9+x_3\\
10x_3+2x_2&=&1-x_1
\end{array}−x1​+2x3​5x1​+x2​10x3​+2x2​​===​5−x2​9+x3​1−x1​​

Writing the augmented matrix of a system of linear equations

Write the augmented matrix for the following system:
x1+6x2−x3−5x4=3x1−x2+4x4=6−3x2−3x3+x4=02x1+3x2−2x3+x4=−1\begin{array}{rcr}
x_1+6x_2-x_3-5x_4&=&3\\
x_1-x_2+4x_4&=& 6\\
-3x_2-3x_3+x_4&=&0\\
2x_1+3x_2-2x_3+x_4&=&-1
\end{array}x1​+6x2​−x3​−5x4​x1​−x2​+4x4​−3x2​−3x3​+x4​2x1​+3x2​−2x3​+x4​​====​360−1​

Practice: Augmented Matrix

Write the augmented matrix that represents this system of linear equations
x1+x2=7−3x32−x1=7x1+x2+x3=0\begin{array}{rcl}
x_1+x_2&=&7-3x_3\\
2-x_1&=&7\\
x_1+x_2+x_3&=&0
\end{array}x1​+x2​2−x1​x1​+x2​+x3​​===​7−3x3​70​

Solving a system of linear equations

Consider the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr}
-2x_1+3x_2+13x_3&=&5\\
-x_1+2x_2+5x_3&=&1\\
4x_1-14x_2-2x_3&=&14\\
\end{array}−2x1​+3x2​+13x3​−x1​+2x2​+5x3​4x1​−14x2​−2x3​​===​5114​
where we wish to solve for the triple (x1, x2, x3) of real numbers that satisfy this system.

Augmented Matrix

Convert the following SLE into an augmented matrix. 
𝑥−𝑦=5𝑧+7x2−z=30=4−x−8y\begin{array}{}
𝑥−𝑦=5𝑧+7\\
\frac{x}{2}-z=3\\
0=4-x-8y
\end{array}x−y=5z+72x​−z=30=4−x−8y​
                                                

Which of the following represents the systems of linear equations corresponding to the following augmented matrices?
[11−302520−4∣630]\left[\begin{array}{l}
1 & 1 & -3  \\
0 &2 & 5 \\
2 & 0 & -4  \\
 \end{array}\right.\left|\begin{array}{l} 6\\3\\0\end{array}\right]
​102​120​−35−4​​630​​

Additional Practice Problems--Systems of Linear Equations

1. Write the augmented matrix that represents this system of linear equations
x1+x2=7−3x32−x1=7x1+x2+x3=0\begin{array}{rcl}
x_1+x_2&=&7-3x_3\\
2-x_1&=&7\\
x_1+x_2+x_3&=&0
\end{array}x1​+x2​2−x1​x1​+x2​+x3​​===​7−3x3​70​
2. Determine all values of a, ba,\ ba, b and ccc if the augmented matrix [2−1001−3]\left[\begin{array}{rr|r}
2&-1&0\\
0&1&-3
\end{array}\right][20​−11​0−3​] represents the following system of linear equations: ax1=bx2(a+c)x2+3=0\begin{array}{rcl}
ax_1&=&bx_2\\
(a+c)x_2+3&=&0
\end{array}ax1​(a+c)x2​+3​==​bx2​0​

Consider a homogeneous system AX=0AX=0AX=0 where AAA is a 2×42\times42×4 matrix. Consider the following statements:
(i) The system is inconsistent.
(ii) X=0X=0X=0 is the unique solution.

Practice: Augmented Matrix (~2016 Test2 #2)

Practice: Augmented Matrix
Answer the following questions about augmented matrices

Augmented Matrix

Answer the following questions about augmented matrices:

Practice: Augmented Matrix

Determine all values of a, ba,\ ba, b and ccc if the augmented matrix [2−1001−3]\left[\begin{array}{rr|r}
2&-1&0\\
0&1&-3
\end{array}\right][20​−11​0−3​] represents the following system of linear equations: 
ax1=bx2(a+c)x2+3=0\begin{array}{rcl}
ax_1&=&bx_2\\
(a+c)x_2+3&=&0
\end{array}ax1​(a+c)x2​+3​==​bx2​0​

Which of the following matrices is a result of performing a single elementary row operation on the augmented matrix [2−2051−43100−24]\left[\begin{array}{rrr|r}
2&-2&0&5\\
1&-4&3&1\\
0&0&-2&4
\end{array}\right]​210​−2−40​03−2​514​​?

Consider a homogeneous system AX=0AX=0AX=0 where AAA is a 2×42\times42×4 matrix. Consider the following statements:
(i) The system is inconsistent.
(ii) X=0X=0X=0 is the unique solution.

Find all the values of kkk such that the system of linear equations corresponding to the augmented matrix  [1201010200k2−9k−3]\left[\begin{array}{rrr|r}
1 & 2 &0 &1  \\

0 & 1  &0&2\\
0&0&k^2-9&k-3
 \end{array}\right]​100​210​00k2−9​12k−3​​has infinitely many solutions.

Find all the values of kkk such that the system of linear equations corresponding to the augmented matrix  [1201010200k2−9k−3]\left[\begin{array}{rrr|r}
1 & 2 &0 &1  \\

0 & 1  &0&2\\
0&0&k^2-9&k-3
 \end{array}\right]​100​210​00k2−9​12k−3​​ has a unique solution.

If * represents any arbitrary number, how many solutions does the linear system with the following augmented matrix have?
[010∗∗003∗∗000a0]\left[\begin{array}{cccc|c} 0 & 1 & 0 & * &* \\ 0 & 0 & 3 & * & *\\ 0 & 0 & 0 & a & 0 \end{array}\right]​000​100​030​∗∗a​∗∗0​​

Augmented matrices

If * represents any arbitrary number, how many solutions does the linear system with the following augmented matrix have?
[010∗∗003∗∗000a0]\left[\begin{array}{cccc|c} 0 & 1 & 0 & * &* \\ 0 & 0 & 3 & * & *\\ 0 & 0 & 0 & a & 0 \end{array}\right]​000​100​030​∗∗a​∗∗0​​

Linear equations and augmented matrices

Suppose that a system of mmmlinear equations in nnnvariables is represented by the augmented matrix [A∣b][A|b][A∣b]. If the system is consistent, which of the following statements is always true?

Linear equations and systems

Given a system of 3 linear equations in 5 variables, if the rank of the coefficient matrix AAAis 3, which of the following statements best describes the system?

Homogenous system with linear equations

A homogeneous system (Ax⃗=0⃗A\vec{x}=\vec{0}Ax=0) with mmmlinear equations in nnnunknowns has exactly one solution. Which of the following statements is always true?

Matrix problem

Given that A=[10−3725411]A=\begin{bmatrix}1 & 0 & -3\\7 & 2 & 5\\4 & 1 & 1\end{bmatrix}A=​174​021​−351​​, which of the following statements is correct?

Augmented Matrix: Gauss-Jordan Elimination

Consider the system of linear equations
x2+4x3=14+2x1−4x4x1−x3−x4+3=02x4−6=0\begin{array}{c}
x_2+4x_3=14+2x_1-4x_4\\
x_1-x_3-x_4+3=0\\
2x_4-6=0
\end{array}x2​+4x3​=14+2x1​−4x4​x1​−x3​−x4​+3=02x4​−6=0​
(a) Write the augmented matrix for the standard form of this system.

123 - FML #1 - 18.1W - e.g. 20

Related Topics

Solving Linear Systems

Augmented Matrix

Consider the following system of linear equations:

{x−2y+4z=22x+y−2z=−13x−y+2z=12x+6y−12z=−6\bcb{ \begin{cases} x - 2y + 4z = 2& \\ 2x + y - 2z = -1 &\\ 3x - y + 2z = 1 &\\ 2x + 6y - 12z = -6& \end{cases}}⎩⎨⎧​x−2y+4z=22x+y−2z=−13x−y+2z=12x+6y−12z=−6​​

a) Construct the augmented matrix A\bcb{A}A corresponding to this system.

b) Solve this system using row operations on A\bcb{A}A. How many solutions are there, if any?

More Solving Linear Systems Questions:

Find size of soln set to nonhom linear system

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 133 - FML 1 - 18.1W e.g. 44.4

Practice Question: Solving a System of Linear Equations (Infinitely Many Solutions)

Wize Concept Clarifier--Gaussian Jordan Elimination

Linear Systems - Mixing

Basics of Systems of Linear Equations

Solve linear systems

123 - FML #1 - 18.1W - e.g. 21.1

Find soln to nonhom linear system

Balancing Chemical Equations

Solving Linear Systems

Solving Linear Systems

Solving Linear Systems

Solving Linear Systems

Solving Linear Systems

19.4F_Final_Builder_Ch_2.13_Algebra_SLE_$\tkco{eg22}$_$\key{Final}$_Builder_$\tkcth{2.13.}\tkcf{22}$_

Interpretation of Linear Systems

Matrix Multiplication

System of Linear Equations

Homogeneous System of Equations

Practice Question: Solving a System of Linear Equations (Unique Solution)

Find size of soln set to nonhom linear system

Practice Question: Solving a System of Linear Equations (Infinitely Many Solutions)

Solve the homogeneous linear system

Solving a system of linear equations

Solving a system of linear equations

Solving a system of linear equations

Practice: Solutions to a System of Equations

Solving a system of linear equations

Practice: Solving a System of Equations

Linear algebra

System of Linear Equations

Solving Linear Systems

Solving Linear Systems

Interpretation of Linear Systems

More Augmented Matrix Questions:

Writing the augmented matrix of a system of linear equations

Writing the augmented matrix of a system of linear equations

Practice: Augmented Matrix

Solving a system of linear equations

Augmented Matrix

Additional Practice Problems--Systems of Linear Equations

Practice: Augmented Matrix (~2016 Test2 #2)

Augmented Matrix

Practice: Augmented Matrix

Augmented matrices

Linear equations and augmented matrices

Linear equations and systems

Homogenous system with linear equations

Matrix problem

Augmented Matrix: Gauss-Jordan Elimination

$\bcb{ \begin{cases} x - 2y + 4z = 2& \\ 2x + y - 2z = -1 &\\ 3x - y + 2z = 1 &\\ 2x + 6y - 12z = -6& \end{cases}}$

a) Construct the augmented matrix $\bcb{A}$ corresponding to this system.

b) Solve this system using row operations on $\bcb{A}$ . How many solutions are there, if any?