Advanced Algebra Notes
Section 3.4: Solve Systems of Linear Equations in Three Variables
linear equation in 3 variables x, y, and z is an equation of the form ax + by + cz = d
A ___________________________
where a, b, and c are not all zero.
system of linear equations in 3 variables:
The following is an example of a ________________________
2x + 5y – z = -7
x – 3y + z = 10
9x + y – 4z = -1
ordered triple (x, y, z) whose
The solution to a system of equation in 3 variables is called an ______________
coordinates make all 3 equations true.
graph of a linear system in 3 variables is a plane in three-dimensional space.
The _______
The intersection of the 3 planes determines the number of solutions.
single point
The planes intersect in a ____________.
line or are the _____________.
same plane
The planes intersect in a ______
(Infinitely Many Solutions)
no common point of intersection. ( No Solution)
The planes have __________________
Steps:
1. Rewrite 2 of the equations in 3 variables as equations in 2 variables using substitution
or elimination.
2. Solve those 2 equations for both variables like you were taught in section 3.2.
3. Once you get the values from step 2, substitute them in to one of the original
equations and solve for the 3rd value. Then write your ordered triple.
** If the variables disappear and you get a false statement -3 = 0, then the system has no
solutions.
** If the variables disappear and you get a true statement 0 = 0, then the system has
infinitely many ordered triple solutions.
Examples:
1.
-y = -2x – 6z – 4
y = 2x + 6z + 4
2x – y + 6z = -4
6x + 4y – 5z = -7
-4x – 2y + 5z = 9
6x + 4(2x + 6z + 4) – 5z = -7
6x + 8x + 24z + 16 – 5z = -7
14x + 19z = - 23
-4x – 2(2x + 6z + 4) + 5z = 9
-4x – 4x – 12z – 8 + 5z = 9
-8x – 7z = 17
14 x 19 z   23
4(14 x 19 z ) (23)4
56 x  76 z   92
8 x  7 z 17
7(8 x  7 z )  (17)7
56 x  49 z 119
27 z  27
8 x  7 z 17
8 x  7(1) 17
8 x  7 17
 8 x  24
x3
y  2x  6z  4
z 1
y  2(3)  6(1)  4
y   664
y 4
 3, 4,1
2.
3x + y – 2z = 10
6x – 2y + z = -2
x + 4y + 3z = 7
3x  y 2 z 10
3x  y  2 z 10
2(6 x  2 y  z )   2(2)
12 x  4 y  2 z   4
15 x  3 y  6
3(6 x  2 y  z )  (2)  3
x  4 y  3z  7
18 x  6 y  3 z 6
x 4 y  3 z  7
17 x 10 y 13
15 x  3 y  6
15(1)  3 y  6
15  3 y  6
 3y   9
10(15 x3 y )  (6)10
150 x  30 y  60
3( 17 x 10 y )  (13)3
51x  30 y  39
99 x  99
x 1
(1,3,  2)
y 3
6x  2 y  z   2
6(1)  2(3)  z   2
66 z  2
z 2
3.
x+y–z=2
3x + 3y – 3z = 8
2x – y + 4z = 7
3( x  y  z )  (2)  3
3 x  3 y  3 z   6
3 x  3 y 3 z  8
3 x  3 y 3 z  8
02

4.
x+y+z=6
x–y+z=6
4x + y + 4z = 24
x y  z 6
x y z 6
x y  z 6
4 x  y  4 z  24
2 x  2 z 12
5 x  5 z  30
5(2 x  2 z )  (12)5
10 x 10 z  60
2(5 x  5 z ) (30)  2
10 x 10 z   60
00
Infinite # of ordered triples