Basics of Systems of Linear Equations

Solving Linear Systems

6 Activities

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​

A)  a⃗=[1t−1]\vec a =
\begin{bmatrix}
  1\\ t\\ -1 
\end{bmatrix}
a=​1t−1​​

B)  b⃗=[5tt2]\vec b =
\begin{bmatrix}
  5t\\ t\\ 2 
\end{bmatrix}
b=​5tt2​​

C)  c⃗=[101]\vec c =
\begin{bmatrix}
  1\\ 0\\ 1 
\end{bmatrix}c=​101​​

D)  d⃗=[−5−12]\vec d =
\begin{bmatrix}
  -5\\ -1\\ 2 
\end{bmatrix}d=​−5−12​​

I don't know

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

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

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.

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​

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