IB Mathematics Analysis and Approaches Topic AHL 1.16—solution of Systems of Linear Equations Revision Notes (HL)

IB Mathematics Analysis and Approaches Topic AHL 1.16—solution of Systems of Linear Equations Revision Notes (HL) | RevisionDojo

Written by official IB examiners

Systems of Linear Equations

You definitely know how to solve systems of two linear equations, for example:

$$\begin{cases}2x+y=6\\5x-y=1\end{cases}$$

However, in AA HL, you need to know the properties of systems of equations with 3 unknowns, in which case the solving becomes a lot more complicated.

Types of Solutions

When solving systems of linear equations, three possible outcomes exist:

Unique solution: The system has exactly one solution.
Infinite solutions: The system has infinitely many solutions.
No solution: The system is inconsistent and has no solution.

Solving Methods

Algebraic Methods

Substitution Method: This method involves expressing one variable in terms of others and substituting it into the remaining equations.
Elimination Method: Also known as addition method, it involves adding or subtracting equations to eliminate variables.
Gaussian Elimination: This method uses elementary row operations to transform the augmented matrix of the system into row echelon form.

Example

Consider the system: $$ \begin{cases} 2x + y - z = 4 \\ x - y + 2z = 5 \\ 3x + 2y - 3z = 3 \end{cases} $$

Substitution Method

Select an equation, and write it in terms of one of the variables. We will select the first equation and write it in terms of $z$ such that we have $2x+y-4=z$.
Then we substitute this value of $z$ into the other two equations.
The second one becomes $x-y+2(2x+y-4)=5$ $\Longrightarrow 5x+y=13$
The third one becomes $3x+2y-3(2x+y-4)=3$ $\Longrightarrow -3x-y=-9$
Hence now we are left with the system of equations $$ \begin{cases} 5x+y=13 \\ 3x+y=9 \end{cases} $$
Now you can repeat the process similarly for these 2 equations to get values of $x$ and $y$ and then solve for the original variable $z$.
In this case, you will get $x=2$ and $y=3$ such that $z=2(2)+3-4=3$

Elimination Method

These methods don't differ from each other much as you will see soon. First, select 2 equations. We will choose the first one and the second one.
Then you want to transform the functions such that when you add them, a variable cancels out. For example, to have a $2x$ in one and a $-2x$ in the other so they will cancel out.
As seen in the first two equations, we have $+y$ and a $-y$ such that we can add them already and get $2x+y-z+x-y+2z=4+5$ $\Longrightarrow 3x+z=9$
Repeat this process for another 2 equations. We will choose the second and the third.
Notice if we transform the second equation by a factor of $2$ then we will have $-2y$ which can cancel the $+2y$ of the third. Hence we can write our second equation as $2x-2y+4z=10$
Hence adding them again you get $5x+z=13$ to give you the system of linear equations $$ \begin{cases} 3x+z=9\\5x+z=13 \end{cases} $$
Now we repeat the process for these 2 equations to solve for $x,y,z$ such that we get $x=2,y=3,z=3$

We will discuss Gaussian elimination separately below

Tip

Substitution and regular elimination get very long very quickly in $3\times3$ systems of linear equations. Gaussian elimination is generally the most efficient method.

Inconsistent Systems

A system is inconsistent when it has no solution. This occurs when the equations contradict each other.

Example

The system: $$ \begin{cases} x + y = 2 \\ x + y = 3 \end{cases} $$ is inconsistent because no values of $x$ and $y$ can simultaneously satisfy both equations.

Geometrically, inconsistent systems represent parallel lines or planes that never intersect.

Inconsistent (no solution) systems will lead to incorrect equalities when solving using any of the above methods.

If you try any method in the example above, you will end up with $-1=0$ which is incorrect.

General Solutions for Infinite Solutions

Nice try, unfortunately this paywall isn't as easy to bypass as you think. Want to help devleop the site? Join the team at https://revisiondojo.com/join-us. exercitation voluptate cillum ullamco excepteur sint officia do tempor Lorem irure minim Lorem elit id voluptate reprehenderit voluptate laboris in nostrud qui non Lorem nostrud laborum culpa sit occaecat reprehenderit

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Duis aute irure dolor in reprehenderit

Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit aut fugit, sed quia consequuntur magni dolores eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci velit.

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Unlock the rest of this chapter with a free account

Systems of Linear Equations

You definitely know how to solve systems of two linear equations, for example:

$$\begin{cases}2x+y=6\\5x-y=1\end{cases}$$

However, in AA HL, you need to know the properties of systems of equations with 3 unknowns, in which case the solving becomes a lot more complicated.

Types of Solutions

When solving systems of linear equations, three possible outcomes exist:

Unique solution: The system has exactly one solution.
Infinite solutions: The system has infinitely many solutions.
No solution: The system is inconsistent and has no solution.

Solving Methods

Algebraic Methods

Substitution Method: This method involves expressing one variable in terms of others and substituting it into the remaining equations.
Elimination Method: Also known as addition method, it involves adding or subtracting equations to eliminate variables.
Gaussian Elimination: This method uses elementary row operations to transform the augmented matrix of the system into row echelon form.

Example

Consider the system: $$ \begin{cases} 2x + y - z = 4 \\ x - y + 2z = 5 \\ 3x + 2y - 3z = 3 \end{cases} $$

Substitution Method

Select an equation, and write it in terms of one of the variables. We will select the first equation and write it in terms of $z$ such that we have $2x+y-4=z$.
Then we substitute this value of $z$ into the other two equations.
The second one becomes $x-y+2(2x+y-4)=5$ $\Longrightarrow 5x+y=13$
The third one becomes $3x+2y-3(2x+y-4)=3$ $\Longrightarrow -3x-y=-9$
Hence now we are left with the system of equations $$ \begin{cases} 5x+y=13 \\ 3x+y=9 \end{cases} $$
Now you can repeat the process similarly for these 2 equations to get values of $x$ and $y$ and then solve for the original variable $z$.
In this case, you will get $x=2$ and $y=3$ such that $z=2(2)+3-4=3$

Elimination Method

These methods don't differ from each other much as you will see soon. First, select 2 equations. We will choose the first one and the second one.
Then you want to transform the functions such that when you add them, a variable cancels out. For example, to have a $2x$ in one and a $-2x$ in the other so they will cancel out.
As seen in the first two equations, we have $+y$ and a $-y$ such that we can add them already and get $2x+y-z+x-y+2z=4+5$ $\Longrightarrow 3x+z=9$
Repeat this process for another 2 equations. We will choose the second and the third.
Notice if we transform the second equation by a factor of $2$ then we will have $-2y$ which can cancel the $+2y$ of the third. Hence we can write our second equation as $2x-2y+4z=10$
Hence adding them again you get $5x+z=13$ to give you the system of linear equations $$ \begin{cases} 3x+z=9\\5x+z=13 \end{cases} $$
Now we repeat the process for these 2 equations to solve for $x,y,z$ such that we get $x=2,y=3,z=3$

We will discuss Gaussian elimination separately below

Tip

Substitution and regular elimination get very long very quickly in $3\times3$ systems of linear equations. Gaussian elimination is generally the most efficient method.

Inconsistent Systems

A system is inconsistent when it has no solution. This occurs when the equations contradict each other.

Example

The system: $$ \begin{cases} x + y = 2 \\ x + y = 3 \end{cases} $$ is inconsistent because no values of $x$ and $y$ can simultaneously satisfy both equations.

Geometrically, inconsistent systems represent parallel lines or planes that never intersect.

Inconsistent (no solution) systems will lead to incorrect equalities when solving using any of the above methods.

If you try any method in the example above, you will end up with $-1=0$ which is incorrect.

General Solutions for Infinite Solutions

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Number and Algebra16 subtopics

Functions16 subtopics

Geometry & Trigonometry18 subtopics

Statistics & Probability14 subtopics

Calculus19 subtopics

IB Mathematics Analysis and Approaches AHL 1.16—solution of Systems of Linear Equations Notes

Number and Algebra16 subtopics

Functions16 subtopics

Geometry & Trigonometry18 subtopics

Statistics & Probability14 subtopics

Calculus19 subtopics

Systems of Linear Equations

Types of Solutions

Solving Methods

Algebraic Methods

Substitution Method

Elimination Method

Inconsistent Systems

General Solutions for Infinite Solutions

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Systems of Linear Equations

Types of Solutions

Solving Methods

Algebraic Methods

Substitution Method

Elimination Method

Inconsistent Systems

General Solutions for Infinite Solutions

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account