Solving Systems of 3 Equations in 3 Variables - Steps & Results

1. 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 • -2x = -2 • x=1 • 8(1) +2y = -4 • 2y = -12 • y = -6 • Substitute x and y into any equation. • x+ y+ z = 4 • 1 + -6 + z = 4 • z= 9 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. 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+16 4(-5z+16)-8z =-20 • -20z +64 -8z = -20 • -28z= -84 • z=3y= -5(3) +16= 1 • 4) substitute the 2 values into one of the original equations • X - 3(1) +2(3) = 11 x= 8 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