Exercise 25 Page 172

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 first 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, 4z was isolated in the third equation. We can now substitute its equivalent expression into the second equation.

x=-4-3y+z -8-7y+4z=13 4z=6+22y

Substitute

(II): 4z= 6+22y

x=-4-3y+z -8-7y+( 6+22y)=13 4z=6+22y

▼

(II): Solve by substitution

RemovePar

(II): Remove parentheses

x=-4-3y+z -8-7y+6+22y=13 4z=6+22y

AddTerms

(II): Add terms

x=-4-3y+z 15y-2=13 4z=6+22y

AddEqn

(II): LHS+2=RHS+2

x=-4-3y+z 15y=15 4z=6+22y

DivEqn

(II): .LHS /15.=.RHS /15.

x=-4-3y+z y=1 4z=6+22y

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

x=-4-3y+z y=1 4z=6+22y

Substitute

(III): y= 1

x=-4-3y+z y=1 4z=6+22( 1)

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(III):Simplify

IdPropMult

(III): Identity Property of Multiplication

x=-4-3y+z y=1 4z=6+22

AddTerms

(III): Add terms

x=-4-3y+z y=1 4z=28

DivEqn

(III):.LHS /4.=.RHS /4.

x=-4-3y+z y=1 z=7

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

x=-4-3y+z y=1 z=7

SubstituteII

(I): y= 1, z= 7

x=-4-3( 1)+ 7 y=1 z=7

▼

(I): Simplify

IdPropMult

(I): Identity Property of Multiplication

x=-4-3+7 y=1 z=7

AddSubTerms

(I): Add and subtract terms

x=0 y=1 z=7

The solution to the system is the point ( 0, 1, 7). This is the singular point at which all three planes intersect. Let's check our solution by substituting the values into the system.

x+3y-z=-4 & (I) 2x-y+2z=13 & (II) 3x-2y-z=-9 & (III)

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

0+3( 1)- 7? =-4 2( 0)- 1+2( 7)? =13 3( 0)-2( 1)- 7? =-9

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

0+3-7? =-4 0-1+14? =13 0-2-7? =-9

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

-4=-4 ✓ 13=13 ✓ -9=-9 ✓

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