Systems of Linear Equations

← Back to MATH1012

Introduction

MATH1012 Chapter 1 Goals:

• Identify linear equations
• Solve systems of linear equations
• Determine how many solutions a system has

A linear equation is an equation whose terms are numbers and variables to the first power.

A solution to a linear equation is a choice of values (numbers) for the variables that satisfies the equation.

Systems of Linear Equations

A system of linear equations (SLE) is a collection of linear equations.

A solution to an SLE must satisfy all the equations.

• We can visualise solutions to SLEs with 2 or 3 variables

  • e.g. using 2D and 3D geometry to plot a line/plane

• Solutions written as: $S=\{(x,y)|x\in \mathbb{R}\}$

  • Variables and set are an example

• An SLE can have:

  • No solutions
  • Unique solution
  • Infinite solutions

Theorem 1.7. (Unit Reader)

Elementary row operations change the system of linear equations but do not alter the solution set.

Keep changing an SLE into a simpler SLE

Read about leading entries in Wk1 lecture slides (page 25)

Summary of Gaussian Elimination Method

• Write system of equations in augmented matrix
• Use Gaussian elimination to convert into row echelon form
• Identify leading entries and basic and non-basic variables

Matrix Notation

When showing working during Gaussian Elimination, show that you have changed the row by writing the way you reduced it ‘overwriting’ the original row.

e.g. R${}_n$ ← R${}_n$ - kR${}_{n-1}$

For the next overwrite, R$n$ will represent what it equalled in the recent matrix and this will simplify your working.

e.g. write R${}_{n+1}$ ← $x$R${}_n$ instead of R${}_{n+1}$ ← $x$(R${}_n$ - 3R${}_{n-1}$)

Using Matrix Form to Solve Simultaneous Equations

• We can use matrix form to solve simultaneous equations with any number of unknown variables (given we have enough information)

• We can write a series of equations as an augmented matrix:
  $\begin{array}{c}{r}_1\\ r_2\\ r_3\end{array}\left[\begin{array}{cccc}{a}_1& b_1& c_1& \alpha \\ a_2& b_2& c_2& \beta \\ a_3& b_3& c_3& \gamma \end{array}\right]$

• We define each equation as a row, e.g. $r_1$ for the first equation, $r_2$ for the second equation, etc.

  • $a_n,b_n,c_n$ are the coefficients for $x,y,z$ for the equation $r_n$
  • The final value in each row is the answer for the given row
    • They can be separated from the coefficients by a dashed line, however I didn’t know how to do that in LaTeX

• Using this form, we can convert it to echelon form to solve for $x,y,z$

• Lets look at converting 3 simultaneous equations into a matrix:
  $\left\{\begin{array}{c}x+y+z=6\\ x+2y+4z=6\\ 2x+3y-3z=20\end{array}\right\}$

  • Becomes:
    $\begin{array}{c}{r}_1\\ r_2\\ r_3\end{array}\left[\begin{array}{cccc}1& 1& 1& 6\\ 1& 2& 4& 6\\ 2& 3& -3& 20\end{array}\right]$

It is ideal to try make the coefficient for $x$ equal to $1$ in the first row, this makes it much easier to convert the matrix into echelon form later on.

Row Echelon Form

• Now that we have a matrix, to solve for $x,y,z$, we can use echelon form:$\begin{array}{c}{r}_1\\ r_2\\ r_3\end{array}\left[\begin{array}{cccc}\#& \#& \#& \#\\ 0& \#& \#& \#\\ 0& 0& \#& \#\end{array}\right]$
• We must convert the numbers below the leading diagonal of our augmented matrix into $0$
  • This gives us a solution for $z$ and an expression in terms of $y$ and $z$
  • Thus it becomes very easy to solve for $x,y,z$
• We can do this by manipulating the rows with arithmetic: $\begin{array}{c}{r}_1\\ r_2-r_1\\ r_3-r_2-r_1\end{array}\left[\begin{array}{cccc}1& 1& 1& 6\\ 0& 1& 3& 0\\ 0& 0& -8& 8\end{array}\right]$
• This rearranged form gives us:
  • $-8z=8$ → $z=-1$
  • $y+3(-1)=0$ → $y=3$
  • $x+(3)+(-1)=6$ → $x=4$
• Thus our solution is: $x=4,y=3$ and $z=-1$

A matrix in row-echelon form will have all rows of zero at the bottom

Reduced Row-Echelon Form

A matrix is in reduced row echelon form if:

• It is in row echelon form
• Each leading entry is equal to 1
• Each leading entry is the only non-zero entry in its column.

Simultaneous Equations with Zero or Infinite Solutions

• You will need to be able to geometrically represent what it means for a series of simultaneous equations to have zero and infinite solutions

• When a series of simultaneous equations have 0 solutions:
  

  Transclude of simulatenousnosolutions

  • Typically it means that we can reduce an equation into the form $ax+by+cz=\lambda$ where $\lambda \ne 0$
    • This equation has no solution
  • Geometrically, it means that the three planes do not intersect all at once at any point

• When a series of simultaneous equations have infinite solutions:
  

  Transclude of simulatenousinfinitesolutions

  • Typically it means one or more equations do not give us any information, i.e. $0x+0y+0z=0$
    • Thus with only two or less simultaneous equations and three unknowns, there are infinite solutions
  • Geometrically it means that the three equations all define the same plane, or the intersection between all three planes is a line

• You may be given a matrix with two undefined values and must state for what values does the matrix have one, infinite or zero solutions

  • You may also have to geometrically represent this information

A row of zeros in an augmented matrix means that row has an identical equation as another row, thus they are geometrically the same plane

Vital Point

• All non-basic variables are free variables
• All basic variables are linear combinations of constants and free variables
  • e.g. $x_1$ is a basic variable defined as $x_1=1-x_2$
  • Leading entries will become basic variables

Always want to achieve row-echelon form and then we can systematically solve the matrix