1. Solving systems of 3 equations in 3 variables
Lesson 29
Solving systems of 3 equations in 3 variables

2. Solving systems of 3 equations
Systems of equations in 3 variables require 3 equations to solve.
The solution to a system of 3 equations in 3 variables is an ordered triple.
It is usually easiest to solve these using elimination

3. To solve a system of 3 equations in 3 variables:
1) use 2 of the 3 equations to eliminate one of the variables
2) use the 3rd equation and one of the first 2 to eliminate the same variable- leaving 2 equations in 2 variables
3) solve the system of equations remaining from step 2
4) substitute the values for the 2 variables in step 3 into one of the original equations to find the 3rd variable.

4. A solution to a system of 3 equations is an ordered triple.
A solution is either a single point or a line.
If the 3 planes do not share a common point, then there is no solution.
There are 3 possible results when solving systems of 3 equations:
1- one solution (planes intersect at exactly 1 point)
2- infinitely many solutions ( same plane or planes intersect in a line)
3- no solutions (parallel planes or planes that each intersect at exactly one of the other planes)

5. Graph of a linear system

6. System of 3 equations with 1 solution
x+ y + z = 4
9x + 3y + z = 0
4x + 2y + z = 1
Eliminate z from 1st & 2nd equation
-x-y-z=-4
9x+3y+z=0
8x +2y = -4
Eliminate z from 2nd & 3rd equation
9x+3y+z = 0
-4x -2y -z = -1
5x +y = -1

7. continue 8x + 2y = -4 8x +2y = -4 5x + y = -1 mult by -2 -10x -2y= 2
Substitute x and y into any equation.
x+ y+ z = 4
z = 4
z= solution is (1,-6,9)

8. Systems with infinitely many solutions
2x + 3y + 4z = 12
-6x -12y -8z = -56
4x + 6y + 8z = 24

9. Systems with no solutions
x + 2y -3z = 4
2x + 4y - 6z = 3
-x + 5y + 3z + 1

10. Types of systems of 3 equations
If you get a false statement when you solve any parts of the system, the system is inconsistent .
The coefficients of the variables are multiples of each other, but the constants are not.

11. consistent 5x - 9y -6z = 11 -5x +9y +6z = -11 2x-4y -3z = 6
1st and 2nd equations are multiples of each other, so they are parallel planes.
Combine the 2nd and 3rd equations and you get -x+y = 1
This is a line and there are an infinite number of solutions to a line

12. Applying substitution to a system of 3 equations
Investigation 3
Applying substitution to a system of 3 equations

13. Solving by substitution
x-3y+2z = 11
-x +4y +3z= 5
2x-2y-4z=2
1) pick an equation and solve it for a variable 1st equation x= 3y-2z+11
2) substitute into the 2nd & 3rd equations
-(3y-2z+11) +4y+3z=5
y +5z =16
2(3y-2z+11)-2y-4z=2
4y-8z=-20
3) rewrite one of the 2 equations you just got for one of the variables and then substitute that into the other equation
y= -5z (-5z+16)-8z =-20
-20z z = -20
-28z= -84
z= y= -5(3) +16= 1
4) substitute the 2 values into one of the original equations
X - 3(1) +2(3) = x= solution (8,1,3)

14. 3-dimensional coordinate system
Points with 3 coordinates are graphed on a 3-dimensional coordinate system.
This is a space that is divided into 8 regions by an x-axis, a y-axis, and a z-axis.
The z-axis is the 3rd axis in a 3-dimensional coordinate plane and is usually drawn as the vertical line.
The y-axis is usually drawn as the horizontal line and the x-axis is drawn as if it is going into the page