Solve x + y = 9
y + z = 7
x – z = 2
That is a dependent system and has an infinite
number of solutions. I'll show you why that is
true, but we start out b y the elimination method:
Begin by writing the equations neatly so that
like terms and equal signs line up vertically:
x + y = 9
y + z = 7
x – z = 2
The idea is to create from this system of
3 equations in the same 3 unknowns to a system
of two equations in the same two unknowns down
to just 1 equation in 1 unknown, which we can
solve for a number and substitute it back in
previous equations to get the other two
unknowns.
The first equation has only 2 unknowns x and
y. We can get another equations in only those
same two unknowns by adding the 2nd and 3rd
equations, so the the z's will cancel out:
y + z = 7
x – z = 2
-------------
x + y = 9
So now we have two equations in the same two
variables and no others.
x + y = 9
x + y = 9
But, strangely enough, they are the same equation,
and so the original system is dependent and there are
infinitely many solutions.
Solve that for y
y = 9 - x
and solve y + z = 7 for z
z = 7 - y
Now to find one solution:
For instance pick x = 1, and substitute in y = 9 - x
y = 9 - 1
y = 8
Substitute that in z = 7 - y
z = 7 - 8
z = -1
So one of many solutions is (x,y,z) = (1,8,-1)
For another solution, I'll arbitrarily pick x = -3, and
substitute that in y = 9 - x
y = 9 - (-3)
y = ( + 3
y = 12
Substitute that in z = 7 - y
z = 7 - 12
z = -5
So another of the many possible solutions is (x,y,z) = (-3,12,-5)
Choose any values for x you like and the use those two equations to
find y and z and find as many solutions to this system as you like.
They'll always check in the original equations.
Edwin
