Exercise 37 Page 172

The given system consists of equations of planes. Let's use the Substitution Method to solve this system of equations. When using this method, it is necessary to isolate a variable. In the third equation, it will be easier 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.

4x-2y+5z=6 3x+3y+8z=4 x=5+5y+3z

(I), (II): x= 5+5y+3z

4( 5+5y+3z)-2y+5z=6 3( 5+5y+3z)+3y+8z=4 x=5+5y+3z

▼

(I), (II): Simplify

Distr

(I): Distribute 4

20+20y+12z-2y+5z=6 3(5+5y+3z)+3y+8z=4 x=5+5y+3z

Distr

(II): Distribute 3

20+20y+12z-2y+5z=6 15+15y+9z+3y+8z=4 x=5+5y+3z

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

20+18y+17z=6 15+18y+17z=4 x=5+5y+3z

SubEqn

(I): LHS-20=RHS-20

18y+17z=-14 15+18y+17z=4 x=5+5y+3z

SubEqn

(I): LHS-18y=RHS-18y

17z=-14-18y 15+18y+17z=4 x=5+5y+3z

This time, 17z was isolated in the first equation. We can now substitute its equivalent expression into the second equation.

17z=-14-18y 15+18y+17z=4 x=5+5y+3z

Substitute

(II): 17z= -14-18y

17z=-14-18y 15+18y+( -14-18y)=4 x=5+5y+3z

RemovePar

(II): Remove parentheses

17z=-14-18y 15+18y-14-18y=4 x=5+5y+3z

AddSubTerms

(II):Add and subtract terms

17z=-14-18y 1≠ 4 x=5+5y+3z

Notice that when we substituted the value of 17z into the second equation, we got a contradiction, 1≠ 4. This means that there is no solution to the system of equations.