Simultaneous Equations 3.15 Practice
To solve a system of equations with 3 variables you need three equations.
Given
3x = y + 4 - 2z
3z = 2x - y
Use each of the five methods below to create the third equation:

multiply the coefficients of the variables and the constant in the first equation, 3𝑥 = 𝑦 + 4 − 2𝑧, by 2.

use the first four numbers of a smart phone passcode 1059, as the coefficients of x, y, and z, and the
constant term in that order when the equation is in the form ax + by +cz = d

write the third equation as 15𝑧 = 10𝑥 − 5𝑦 − 20.

Add the first equation and the second equation to form the third equation.

Use the numbers 4, -3, -3, 4 as the coefficients of x, y, and z, and the constant term in that order when
the equation is in the form ax + by +cz = d
Solve each set of three equations and give a geometric interpretation of the solution. Write a general
statement about the solution for each set.
As you write your report, include the equations you have used, as well as relevant calculations and/or
diagrams. Use appropriate mathematical statements to communicate your findings.
SOLUTIONS
3x - y + 2z = 4
a. 3z = 2x - y
6x - 2y + 4z = 8
é
ê
ê
ê
ë
é
ê
2 ´ eqn1- 3´ eqn2 ê
ê
2eqn1- eqn3
ë
3 -1 2 4 ù
ú
2 -1 -3 0 ú
6 -2 4 8 úû
3 -1 2 4 ù
ú
0 1 13 8 ú
0 0 0 0 úû
0z = 0 has infinite solutions. The equations are dependent and the three planes meet in a line
The third equation is double the first equation, including the constant so the third plane is exactly the
same as the first plane. The first and third planes meet with infinite solns. All three planes intersect in an
infinite number of points along a line.
In general when one equation is a multiple of a second equation – inculuding the constant, then the two
planes will sit on top of each other and the three planes will meet in a line giving infinite solutions.
3x - y + 2z = 4
b.
3z = 2x - y
x + 5z = 9
é
ê
ê
ê
ë
é
ê
2 ´ eqn1- 3´ eqn2 ê
eqn1- 3´ eqn3 êë
3 -1 2 4 ù
ú
2 -1 -3 0 ú
1 0 5 9 úû
3 -1 2
4 ù
ú
0 1 13
8 ú
0 -1 -13 -23 úû
eqn2 + eqn3
é 3 -1 2
4 ù
ê
ú
ê 0 1 13 8 ú
ê 0 0 0 -15 ú
ë
û
0 z = -15 is impossible to solve. The equations are inconsistent.
The third plane is parallel to the line of intersection of the other two planes and there is no point of
intersection.
In general whenever a solution is of the type 0z = a, and none of the planes are parallel to each other then
the third plane will always be parallel to the intersection of the other two planes and there will never be
any solutions to the equations.
OR
b.
3x - y + 2z = 4
x 's
3a + 2b = 1
3z = 2x - y
y's
-a - b = 0
x + 5z = 9
b = -a
3a + 2(-a) = 1
a =1
b = -1
The 1st eqn minus the 2nd eqn equal the 3rd eqn, but not for the constant.
The third plane is parallel to the line of intersection of the other two planes and there is no point of
intersection.
In general when a linear combination of two equations equal the third equation but not for the constant
there will never be any solutions to the equations as the third plane will be parallel to the intersection of
the other two planes.
c.
3x - y + 2z = 4
3z = 2x - y
15z = 10x - 5y - 20
é 3
ê
ê 2
ê 10
ë
é 3
ê
2 ´ eqn1- 3´ eqn2 ê 0
10 ´ eqn1- 3´ eqn3 êë 0
5 ´ eqn2 - eqn3
-1 2
4 ù
ú
-1 -3 0 ú
-5 -15 20 úû
-1 2
4 ù
ú
1 13 8 ú
5 65 -20 úû
é 3 -1 2 4 ù
ê
ú
ê 0 1 13 8 ú
ê 0 0 0 60 ú
ë
û
0z = 60 is impossible to solve. The equations are inconsistent
The 3rd eqn is five times the 2nd eqn, but not for the constant so the two planes formed by these equations
are parallel and there is no point of intersection.
In general when two of the equations are multiples of each other but not for the constant they give parallel
planes and their will never be any points of intersection or solutions to the equation.
d.
3x - y + 2z = 4
2x - y - 3z = 0
5x - 2y - z = 4
é
ê
ê
ê
ë
é
ê
2 ´ eqn1- 3´ eqn2 ê
5´ eqn1- 3´ eqn3 êë
3 -1 2 4 ù
ú
2 -1 -3 0 ú
5 -2 -1 4 úû
3 -1 2 4 ù
ú
0 1 13 8 ú
0 1 13 8 úû
eqn2 - eqn3
é 3 -1 2 4 ù
ú
ê
ê 0 1 13 8 ú
ê 0 0 0 0 ú
û
ë
0z = 0 has infinite solutions. The equations are dependent and the three planes intersect in a line.
In general when the solution is of the type 0z = 0 and none of the equations is a multiple of the others
then there will always be infinite solutions with the three planes meeting in a line.
OR
d.
3x - y + 2z = 4
2x - y - 3z = 0
5x - 2y - z = 4
x 's
3a + 2b = 5
y's
-a - b = -2
a = 2-b
3(2 - b) + 2b = 5
6 - 3b + 2b = 5
b =1
a =1
The 1st eqn plus the 2nd eqn give the third equation, including the constant so the three planes formed by
these equations intersect in a line which gives infinite solutions. The equations are dependent.
In general when a linear combination of two equations equal the third equation including the constant the
system of equations are dependent and the planes will always meet in a line giving infinite solutions
e.
3x - y + 2z = 4
2x - y - 3z = 0
4x - 3y - 3z = 4
é
ê
ê
ê
ë
é
ê
2 ´ eqn1- 3´ eqn2 ê
4 ´ eqn1- 3´ eqn3 êë
5 ´ eqn2 - eqn3
3 -1 2 4 ù
ú
2 -1 -3 0 ú
4 -3 -3 4 úû
3 -1 2 4 ù
ú
0 1 13 8 ú
0 5 17 4 úû
é 3 -1 2 4 ù
ê
ú
ê 0 1 13 8 ú
ê 0 0 48 36 ú
ë
û
36
z=
= 0.75
48
y +13(0.75) = 8
y = -1.75
3x - (-1.75) + 2(0.75) = 4
3x = 0.75
x = 0.25
The three equations have a unique solution. The three planes meet in a single point with x = 0.25, y = -1.75,
z = 0.75
In general when three equations have a unique solution the three planes will always meet in a single point.