Exercise 43 Page 218

The given system consists of equations of planes. Notice that the coefficient of y in the first equation is the additive inverse of the coefficient of y in the third equation; they will add to be 0. Let's use the Elimination Method to find a solution to this system. 2x - y +3z=-1 & (I) x+2y-4z=-1 & (II) y-2z=0 & (III) We can start by adding the third equation to the first equation to eliminate the y-terms. Having eliminated the y-variable from the first equation, we can continue by creating additive inverse coefficients for y in the second and third equations. Then, we can add or subtract these equations to eliminate y from the second equation.

2x+z=-1 x+2y-4z=-1 y-2z=0

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Solve by elimination

MultEqn

(III): LHS * 2=RHS* 2

2x+z=-1 x+2y-4z=-1 2(y-2z)=2(0)

Distr

(III): Distribute 2

2x+z=-1 x+2y-4z=-1 2y-2(2z)=2(0)

Multiply

(III): Multiply

2x+z=-1 x+2y-4z=-1 2y-4z=2(0)

ZeroPropMult

(III): Zero Property of Multiplication

2x+z=-1 x+2y-4z=-1 2y-4z=0

SysEqnSub

(II): Subtract (III)

2x+z=-1 x+2y-4z-( 2y-4z)=-1-( 0) 2y-4z=0

RemovePar

(II): Remove parentheses

2x+z=-1 x+2y-4z-2y+4z=-1-0 2y-4z=0

AddSubTerms

(II): Add and subtract terms

2x+z=-1 x=-1 2y-4z=0

Now that we know that x=-1, we can substitute it into the first equation to find the value of z.

2x+z=-1 x=-1 2y-4z=0

Substitute

(I): x= -1

2( -1)+z=-1 x=-1 2y-4z=0

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(I): Solve for z

MultPropNegOne

(I): Multiplication Property of -1

-2+z=-1 x=-1 2y-4z=0

AddEqn

(I): LHS+2=RHS+2

-2+z+2=-1+2 x=-1 2y-4z=0

AddTerms

(I): Add terms

z=1 x=-1 2y-4z=0

The value of z is 1. Let's substitute this value into the third equation to find y.

z=1 x=-1 2y-4z=0

Substitute

(III): z= 1

z=1 x=-1 2y-4(1)=0

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(III): Solve for y

IdPropMult

(III): Identity Property of Multiplication

z=1 x=-1 2y-4=0

AddEqn

(III): LHS+4=RHS+4

z=1 x=-1 2y-4+4=0+4

AddTerms

(III): Add terms

z=1 x=-1 2y=4

DivEqn

(III): .LHS /2.=.RHS /2.

z=1 x=-1 2y/2=4/2

CalcQuot

(III): Calculate quotient

z=1 x=-1 y=2

The solution to the system is x=-1, y=2, and z=1. This is the singular point at which all three planes intersect.