Solving linear systems in three unknowns

A system of equations is a group of $2$2 or more equations, we call it a system when maybe the equations are related by context or purpose, kind of like that the system is working together.  The individual equations can be linear, quadratic or indeed take any form but for now we will concentrate on just the linear ones.  There are two useful features about systems of equations, the first is the solution to the system and the second relates to regions within the system.

Solution to the systems

The solution to a system is the point where all the equations are satisfied.  When we had $2$2 linear equations in $2$2 unknowns the solution was the intersection of the two lines because at this point both lines are satisfied.

An equation in $3$3 unknowns takes us up a dimension.  No longer are we working in $2$2 dimensions of the $x$x and $y$y axis, but in $3$3 dimensions with the $x,y$x,y and $z$z axis. This is the $3$3 dimensional cartesian plane.  You can see the $x$x and $y$y axis in red and green, and the $3$3rd dimension is $z$z, which is in blue.

The line $y=x$y=x appears in the $3$3 dimensions in what we call a plane.  It is a surface.  It is still taking on the familiar $y=x$y=x shape, but the third dimension gives us this shape.  It is similar to imaging holding up a piece of paper.  

The equation $x+y+z=1$x+y+z=1, produces another plane surface. 

 

The following applet will allow you to create different planes using the sliders for different values of $x,y,z$x,y,z.  

So now we know what an equation in $3$3 dimensions produces.  

Let's have a brief look at the behaviour of planes.

 

Planes can be parallel

For example here we have pictured $x+y+z=1,x+y+z=-3$x+y+z=1,x+y+z=−3 and $x+y+z=5$x+y+z=5

Planes can intersect 

Here we have pictured $x+y+z=1$x+y+z=1 and $-2x+y+5z=1$−2x+y+5z=1.

We can see here that we don't have a single point of intersection, but we have a line. 

 

With three planes intersecting we end up with a point. It's at this point that the solution to the system of $3$3 equations and $3$3 unknowns provides us.

This image shows the line that was created by the intersection of $x+y+z=1$x+y+z=1 and $-2x+y+5z=1$−2x+y+5z=1 and also the plane created by $3x+0.5y-0.5z=1$3x+0.5y−0.5z=1.  We can see that these intersect at just the one point. The place where the line passes through the plane.

 

Here is the original image of the $3$3 planes.

The point of intersection for this set is $(0.29,0.5,0.21)$(0.29,0.5,0.21).

 

Solving systems of equations in three unknowns.

Now we know what we are finding, (the intersection of the planes), then we need to set about doing it.

The good news is, we use all the tools we have already gained solving systems in $2$2 unknowns.  We can use substitution, elimination or like I did above a graphical approach using technology. 

Let's do one together.

Solve the system of equations

$2x+y-z$2x+y−z	$=$=	$9$9	$(1)$(1)
$x-3y+z$x−3y+z	$=$=	$-2$−2	$(2)$(2)
$-x+y+3z$−x+y+3z	$=$=	$-8$−8	$(3)$(3)

To solve this let's use elimination, first on equations $1$1 and $2$2

$2x+y-z$2x+y−z	$=$=	$9$9	$(1)$(1)
$x-3y+z$x−3y+z	$=$=	$-2$−2	$(2)$(2)
$3x-2y$3x−2y	$=$=	$7$7	$(1)+(2)->(4)$(1)+(2)−>(4)

and a similar operation with equations $1$1 and $3$3.

$2x+y-z$2x+y−z	$=$=	$9$9	$(1)$(1)
$-x+y+3z$−x+y+3z	$=$=	$-8$−8	$(3)$(3)
$6x+3y-3z$6x+3y−3z	$=$=	$27$27	$3\times(1)->(5)$3×(1)−>(5)
$5x+4y$5x+4y	$=$=	$19$19	$(5)+(3)->(6)$(5)+(3)−>(6)

Now we have this smaller system in $2$2 unknowns that we can solve for $x$x and $y$y.

$3x-2y$3x−2y	$=$=	$7$7	$(4)$(4)
$5x+4y$5x+4y	$=$=	$19$19	$(6)$(6)
$11x$11x	$=$=	$33$33	$(6)+2\times(4)$(6)+2×(4)
$x$x	$=$=	$3$3

So $x=3$x=3, which means that 

$3x-2y$3x−2y	$=$=	$7$7
$3\cdot3-2y$3·3−2y	$=$=	$7$7
$9-2y$9−2y	$=$=	$7$7
$-2y$−2y	$=$=	$-2$−2
$y$y	$=$=	$1$1

Substituting back into (1), $2x+y-z=9$2x+y−z=9 we discover that $2\times(3)+1-z=9$2×(3)+1−z=9, which means that $z=-2$z=−2

So our final solution is that 

$x=3$x=3, $y=1$y=1 and $z=-2$z=−2.  We can write this as an ordered triple of $(3,1,-2)$(3,1,−2).

The great thing about systems of equations is that you can always substitute back in and check your answer. 

 

Is this the solution?

We can check that a point (in $3$3 dimensions we call this point an ordered triple) is a solution by substituting into each of the equations.  If the values for $x,y$x,y and $z$z satisfy ALL three equations, then the ordered triple is a solution. 

 

Worked Example 1

 

Types of Solutions

No Solutions

I already eluded to a special case of systems of equations in 3 unknowns, when they are all parallel.  If the solution to a system is the point (or line) that occurs on their intersection, then what happens when there are no intersections?  

In the case of parallel lines, the system is what we call inconsistent.  When you try to solve a system that is parallel, you end up with what we call contradictions.  Take our parallel line example from earlier.  

For example here we have pictured $x+y+z=1,x+y+z=-3$x+y+z=1,x+y+z=−3 and $x+y+z=5$x+y+z=5

See how $x+y+z$x+y+z is $1$1, but it is also $-3$−3 and $5$5.  Clearly, there is not a set of numbers that when we add them together we can have three different answers.  So this is the contradiction, and it is what indicates parallel lines. 

 

Infinite Solutions

Infinite solutions result when one of the equations ends up being (through some kind of algebraic manipulation) the same as one of the other equations.  This means we really only have two unique equations in three unknowns so we are not actually able to solve it. . 

For example, this system has infinite solutions as equation $(1)$(1) and equation $(3)$(3) are actually a multiple of each other. 

$3x-2y+z=5$3x−2y+z=5 $1$1

$5x+y+z=-2$5x+y+z=−2 $2$2

$-3x+2y-z=-5$−3x+2y−z=−5 $3$3

If a system has infinite solutions we say that the system is dependent.  

One Solution

If a system has one solution, than we say that the system is independent.  

 

 

Final Summary

 

 

More Examples - try these ones out

Example 2

Example 3