Wize High School Algebra II Textbook (Common Core) > Systems of Linear Equations (Extension Topic)

Basics of Systems of Linear Equations

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Basics of Systems of Linear Equations
Linear Equations
A linear equation with unknowns x1,x2,…,xn\colorOne{x_1,x_2,\dots,x_n}x1​,x2​,…,xn​ is an equation written in the form:
𝑎1𝑥1+𝑎2𝑥2+⋯+𝑎𝑛𝑥𝑛=𝑏\boxed{\quad
𝑎_1\colorOne{𝑥_1}+𝑎_2\colorOne{𝑥_2}+⋯+𝑎_𝑛\colorOne{𝑥_𝑛}=𝑏
\quad}a1​x1​+a2​x2​+⋯+an​xn​=b​
Notes
Each term is of degree 1   ⟹  \implies⟹ a coefficient ai∈Ra_i \in \realsai​∈R times a single variable/unknown
aia_iai​ cannot all be 0

Examples
linear equation with 2 unknowns:  ln⁡(3)x−1πy=1\ln(3)x-\dfrac{1}{\pi}y=1ln(3)x−π1​y=1
linear equation with 4 unknowns: −4𝑥1+5𝑥2−x3+2𝑥4=e2-4𝑥_1 + 5𝑥_2 - x_3 + 2𝑥_4 = e^2−4x1​+5x2​−x3​+2x4​=e2
non-linear equation with 2 unknowns:  sin⁡(3x)−y2=1\sin(3x)-y^2=1sin(3x)−y2=1

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System of Linear Equations
A system of linear equations (SLE) or linear system is a collection of linear equations with the same variables x1,x2,…,xn\colorOne{x_1,x_2,\dots,x_n}x1​,x2​,…,xn​:
m equations {a11x1+a12x2+⋯+a1nxn=b1a21x1+a22x2+⋯+a2nxn=b2⋮am1x1+am2x2+⋯+amnxn=bmm \text{ equations }
\left\{
\begin{array}{ccccccccl}
  a_{11}\colorOne{\colorOne{x_1}}&+&a_{12}\colorOne{\colorOne{x_2}}&+&\cdots&+&a_{1n}\colorOne{\colorOne{x_n}} &=& b_1\\
  a_{21}\colorOne{x_1}&+&a_{22}\colorOne{x_2}&+&\cdots&+&a_{2n}\colorOne{x_n} &=& b_2\\
  &&&\vdots&&\\
    a_{m1}\colorOne{x_1}&+&a_{m2}\colorOne{x_2}&+&\cdots&+&a_{mn}\colorOne{x_n} &=& b_m\\
\end{array}
\right.m equations ⎩⎨⎧​a11​x1​a21​x1​am1​x1​​+++​a12​x2​a22​x2​am2​x2​​++⋮+​⋯⋯⋯​+++​a1n​xn​a2n​xn​amn​xn​​===​b1​b2​bm​​

A system with mmm equations (rows) and nnn unknowns (terms/columns) is of size m×nm\times nm×n.

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Solutions to Linear Systems
A solution to an m×nm\times nm×n  SLE is a vector x⃗=⟨x1,x2,…,xn⟩\vec x = \lang x_1, x_2, \dots, x_n\rangx=⟨x1​,x2​,…,xn​⟩ that satisfies all mmm equations.
Two SLEs are equivalent if they have the same solution.
Example
x⃗=[xy]=[1−2]\vec x 
= 
\begin{bmatrix}
  x\\ y
\end{bmatrix}
= 
\begin{bmatrix}
  1\\ -2
\end{bmatrix}x=[xy​]=[1−2​] is a solution to the following equivalent SLEs:
  x+y=−12x−y=4\begin{array}{rcl}
x+y&=&-1\\
2x-y&=&4
\end{array}x+y2x−y​==​−14​   and   2x+2y=−22x−y=4\begin{array}{rcl}
2x+2y&=&-2\\
2x-y&=&4
\end{array}2x+2y2x−y​==​−24​
Family of Solutions
A k\colorFour{\bm{k}}k-parameter family of solutions to a SLE is a set of solutions which contains kkk parameters (real number variables).
A family of solutions consists of infinitely many solutions (but not every vector is a solution!)
Example
x⃗=[1t2−ss]\vec x = 
\begin{bmatrix}
  1\\ t\\ 2-s\\ s
\end{bmatrix}x=​1t2−ss​​ is a 2-parameter family of solutions (with parameters s, t∈Rs,\ t \in \realss, t∈R) to the SLE:
x1+x3+x4=3x1−x3−x4=−1\begin{aligned}
x_1 +x_3+x_4&=3\\
x_1 -x_3-x_4 &=-1
\end{aligned}x1​+x3​+x4​x1​−x3​−x4​​=3=−1​
Wize Tip
We can rewrite the general solution to clearly see that it is a 2-parameter family of solutions:
x⃗=[1020]+t[0100]+s[00−11]\vec x = 
\begin{bmatrix}
  1\\ 0\\ 2\\ 0
\end{bmatrix}
+
t\begin{bmatrix}
  0\\ 1\\ 0\\ 0
\end{bmatrix}
+s\begin{bmatrix}
  0\\ 0\\ -1\\ 1
\end{bmatrix}x=​1020​​+t​0100​​+s​00−11​​ 

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Example: Linear Equations
1. Determine whether the following are linear equations in the variables x, y, zx,\ y,\ zx, y, z.
A)  5x−y2=3−0.5z\sqrt{5}x-\frac{y}{2}=3-0.5z5​x−2y​=3−0.5z

Linear. (Can be rearranged to the standard form)
The variables x, y, zx,\ y,\ zx, y, z are all linear with constant coefficients in front.

B)  2𝑥2−sin⁡𝑦+1𝑧=02𝑥^2−\sin𝑦+\frac1𝑧=02x2−siny+z1​=0

Non-linear.
x2x^2x2 is quadratic, sin⁡ y\sin\ ysin y is a trigonometric function, 1z\frac{1}{z}z1​ is a reciprocal function.

C)  sin⁡(40)𝑥+24𝑧=−13\sin(40)𝑥+2^4𝑧=−\frac{1}{3}sin(40)x+24z=−31​

Linear.
x, zx,\ zx, z are linear with constant coefficients in front, and the coefficient in front of yyy is 0.

D)  sin⁡(0)x+(e0−1)z=0\sin(0)x + (e^0 - 1)z = 0sin(0)x+(e0−1)z=0

Non-linear.
The coefficients for each of the variables x, y, zx,\ y,\ zx, y, z is 0 (yyy does not appear,  sin⁡(0)=0\sin(0)=0sin(0)=0, and e0−1=0e^0 -1 = 0e0−1=0).
This equation just says 0=00=00=0, and since there are no variables, it is not linear.

Which of the following are linear equations in the variables p, q, rp,\ q,\ rp, q, r? [Select all that apply]

A) xp+y𝑞2+z𝑟3=3 xp + y𝑞^2 + z𝑟^3 = 3xp+yq2+zr3=3

B) 𝑝𝑥−𝑞𝑦−𝑧=0 \sqrt{𝑝𝑥} − 𝑞𝑦 − 𝑧 = 0px​−qy−z=0

C) 𝑥2p+yq − 𝑟𝑧=10𝑥^{2}p+yq\ −\ \dfrac{𝑟}{𝑧}=10x2p+yq − zr​=10

D) 𝑝𝑥=0𝑝𝑥=0px=0

I don't know

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