Solve. You will find a system with an infinite number of solutions, with no solu

Question

Solve. You will find a system with an infinite number of solutions, with no solution, or with a unique solution. (Enter your answers as a comma-separated list. If there are infinitely many solutions, enter a parametric solution using t and s as the parameters. If there is no solution, enter NONE.)

3x ? 24y ? 24z = 36

-4x + 32y + 32z = -48

Solve. (Enter your answers as a comma-separated list. If there are infinitely many solutions, enter a parametric solution. If there is no solution, enter NONE.)

Explanation / Answer

1)

3x ? 24y ? 24z = 36 => 3(x - 8y -8z) =36 => x - 8y -8z=12 ------------------>(a)

-4x + 32y + 32z = -48 => -4(x -8y -8z) =-48 =>x - 8y -8z =12---------------->(b)

(a) =(b)

so there exists infinate solutions

x - 8y -8z=12 => x =12+8y +8z

let y=t , z=s

=>x =12+8t +8s

therefore the solution is (x,y,z)=(12+8t +8s , t ,s)

=====================================================

2)

40x +5y =20 =>5(8x+y) =20 =>8x+y =4

9x-3y=21=> 3(3x-y)=21 => 3x-y =7

8x+y+3x-y =4+7

=>11x =11

=>.x=1

3x-y =7 , x=1

=>3*1 -y =7

=>3-y=7

=>y=-4

substitute x=1, y =-4 in 22x+11y =-22

=>22*1 +11*(-4) =-22

=>22-44=-22

=>-22 =-22 ----------->true

therefore the solution is (x,y)=(1,-4)

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