MAC 1140
Sect 8.2
Systems of Linear Equations in Three Variables
A linear equation in three variables x, y, and z is an equation of the form
Ax + By + Cz = D ,
where A, B, C, and D are real numbers with A, B, and C not all equal to zero. (For example, x + 2y + z = 6)
A solution to a linear equation in three variables is an ordered triple of real numbers in the form (x, y, z)
that satisfies the equation.
Are the following solutions to the equation x + 2y + z = 6?
a) (1, 0, 5)
b) (2, -3, 10)
YOU TRY: c) (0, 2, 8)
d) (3, -2, 7)
Recall: An independent system is a system whose solution is a single ordered triple.
A dependent system is a system which has infinitely many solutions.
An inconsistent system is a system that does not have a single point in all three planes. Thus, its
solution is the empty set.
Solve each system of equations:
e) 3x – y + 2z = 14
x+y–z=0
2x – y + 3z = 18
f) 4x – 2y + z = 13
3x – y + 2z = 13
x + 3y – 3z = -10
h) 2x – 6y + 4z = 8
3x – 9y + 6z = 12
5x – 15y + 10z = 20
g) x + 2y + z = 4
2x – y – z = 3
i) -2x + y – 3z = 6
4x – y + z = 2
2x – y + 3z = 1
YOU TRY:
j) 2x + y – 2z = -15
4x – 2y + z = 15
X + 3y + 2z = -5
k) -2x + 2y – z = 4
2x – y + z = 1
(074)