Exercise 62 Page 416

The given system consists of equations of planes. When using the Substitution Method to solve a system of equations, it is necessary to isolate a variable. In the second equation it will be easiest to isolate x. With a variable isolated in one of the equations, we can substitute its equivalent expression into the remaining equations. In the final step of the simplification of these substitutions, our goal is to have yet another variable isolated. This time, - 3y was isolated in the third equation. We can now substitute its equivalent expression into the first equation.

- 3y=8+11z x=- 1-y-4z - 3y=11+14z

Substitute

(I): - 3y= 11 + 14z

11 +14z=8+11z x=- 1-y-4z - 3y=11+14z

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(I): Solve by substitution

SubEqn

(I): LHS-11=RHS-11

14z=- 3+11z x=- 1-y-4z - 3y=11+14z

SubEqn

(I): LHS-11z=RHS-11z

3z=- 3 x=- 1-y-4z - 3y=11+14z

DivEqn

(I): .LHS /3.=.RHS /3.

z=- 1 x=- 1-y-4z - 3y=11+14z

The value of z is - 1. Substituting - 1 for z into the third equation, we can find the value of y.

z=- 1 x=- 1-y-4z - 3y=11+14z

Substitute

(III): z= - 1

z=- 1 x=- 1-y-4z - 3y=11+14( - 1)

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(III): Solve by substitution

MultPropNegOne

(III): Multiplication Property of -1

z=- 1 x=- 1-y-4z - 3y=11-14

SubTerm

(III): Subtract term

z=- 1 x=- 1-y-4z - 3y=- 3

DivEqn

(III): .LHS /(- 3).=.RHS /(- 3).

z=- 1 x=- 1-y-4z y=1

Now that we know the values of y and z, we are able to find the value of x.

z=- 1 x=- 1-y-4z y=1

SubstituteII

(II): y= 1, z= - 1

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

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(II): Simplify

MultPropNegOne

(II): Multiplication Property of -1

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

AddSubTerms

(II): Add and subtract terms

z=- 1 x=2 y=1

The solution to the system is the point ( 2, 1, - 1). This is the singular point at which all three planes intersect. Let's check our solution by substituting the values into the system.

2x-y-3z=6 & (I) x+y+4z=- 1 & (II) 3x-2z=8 & (III)

(I), (II), (III): Substitute values

2( 2)- 1-3( - 1)? =6 & (I) 2+ 1+4( - 1)? =- 1 & (II) 3( 2)-2( - 1)? =8 & (III)

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(II): Simplify

(I), (II), (III): Multiplication Property of -1

2(2)-1+3? =6 & (I) 2+1-4? =- 1 & (II) 3(2)+2? =8 & (III)

(I), (II), (III): Multiply

4-1+3? =6 & (I) 2+1-4? =- 1 & (II) 6+2? =8 & (III)

(I), (II), (III): Add and subtract terms

6=6 - 1=- 1 8=8

Since the substitution of our answers into the given equations resulted in three identities, we know that our solution is correct.