Systems of Equations
Summary
Independent
• The equations of a linear system are
independent if none of the equations can be
derived algebraically from the others.
Independence
• When the equations are independent, each
equation contains new information about the
variables, and removing any of the equations
increases the size of the solution set.
Is this set of equations independent or
dependent?
x+ y+z = 2
2x - 3y + 2z = 5
3x - 2y + 3z = 7
Is this set of equations independent or
dependent?
x+ y+z = 2
2x - 3y + 2z = 5
3x - 2y + 3z = 7
Give a geometrical explanation of this
system of liner equations.
x+ y+z = 2
2x - 3y + 2z = 5
3x - 2y + 3z = 7
The three planes intersect on a line.
There are an infinite number of solutions
x+ y+z = 2
2x - 3y + 2z = 5
3x - 2y + 3z = 7
Consistent
• A linear system is consistent if it has a
solution, and inconsistent otherwise.
• When the system is inconsistent, it is possible
to derive a contradiction from the equations,
that may always be rewritten such as the
statement 0 = 1.
• Write a system of equations that describe 3
parallel planes.
• Is the system dependent or independent?
• Is the system consistent or inconsistent?
Example
• Independent and inconsistent
x + 2y + 3z = 4
2x + 4y + 6z = 10
2x + 4y + 6z = 12
• Write a system of equations that describe 2
parallel planes cut by a third plane.
• Is the system dependent or independent?
• Is the system consistent or inconsistent?
Example
• Independent and inconsistent
x + 2y + 3z = 4
2x + 4y + 6z = 10
2x + 3y + 5z = 3
• Write a system of equations that describe 3
planes that form a triangular prism.
• Is the system dependent or independent?
• Is the system consistent or inconsistent?
Example
• Independent and inconsistent
x + 2y + 3z = 4
2x + 4y + 6z = 10
3x + 6y + 9z = 3
• Write a system of equations that describe 3
planes that intersect on a line.
• Is the system dependent or independent?
• Is the system consistent or inconsistent?
Example
• Dependent and consistent with an infinite
number of solutions
x + 2y + 3z = 4
2x + 4y + 6z = 10
3x + 6y + 9z = 14