Theoretically, a system of this type can be solved by graphing. However, the graph of
a linear equation with three variables is a plane,not a line. Since visualizing a plane
requires three-dimensional graphing, the method of graphing is not practical with
these systems. However, it does illustrate the number of solutions possible for such
systems, as shown in FIGURE 7.
226 CHAPTER 4 Systems of Linear Equations
System of linear equations
in three variablesA single solutionI P
II
III(d) (e) (f) (g)
FIGURE 7Points of a line in commonILII
IIIAll points in commonI, II, IIIIIIIIINo points in common No points in commonIIII
II
III III
LNo points in common No points in commonI, II
III(a) (b) (c)OBJECTIVES A solution of an equation in three variables, such as
Linear equation in three variablesis called an ordered tripleand is written For example, the ordered triple
is a solution of the preceding equation, because
is a true statement. Verify that another solution of this equation is
We now extend the term linear equationto equations of the form
where not all the coefficients A, B, C,... , Dequal 0. For example,
and
are linear equations, the first with three variables and the second with four.
OBJECTIVE 1 Understand the geometry of systems of three equations in
three variables.Consider the solution of a system such as the following.
2 x- 3 y+ 2 z= 3
x+ 7 y- 3 z=- 14
4 x+ 8 y+ z= 2
2 x+ 3 y- 5 z= 7 x- 2 y- z+ 3 w= 8
Ax+By+Cz+ Á+ Dw=K,
1 10, -3, 7 2.
2102 + 3112 - 1 - 12 = 4
10 , 1 , - 12
1 x, y, z 2.
2 x+ 3 y-z= 4,
Systems of Linear Equations in Three Variables
4.2
1 Understand the
geometry of
systems of three
equations in three
variables.
2 Solve linear systems
(with three
equations and three
variables) by
elimination.
3 Solve linear systems
(with three
equations and three
variables) in which
some of the
equations have
missing terms.
4 Solve special
systems.