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

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

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.

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)

Graph of a linear system

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

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 1 + -6 + z = 4 z= 9 solution is (1,-6,9)

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

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

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.

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

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

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=3 y= -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)

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