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Use substitution to solve the following system of linear equations and fill in the following blanks:
x + y + z = -6
x - 6y - 7z = -29
-7y - 5z = 4
In the first step, select Equation 1 and find an expression for variable X in terms of other variables:
x = - y -z - 6
Then substitute X in Equation 2. The result is the following new equation:
___y+___ z = -23
In the last step, using back-substitution, the solution for this system is:
x=
y=
Z=

Respuesta :

Answer:

[tex]x=-4[/tex]

[tex]y=3[/tex]

[tex]z=-5[/tex]

Step-by-step explanation:

Given:

[tex]x+y+z=-6[/tex]

[tex]x-6y-7z=-29[/tex]

[tex]-7y-5z=4[/tex]

Solve for [tex]x[/tex] in the 1st equation:

[tex]x+y+z=-6[/tex]

[tex]x+y=-z-6[/tex]

[tex]x=-y-z-6[/tex]

Substitute the value of [tex]x[/tex] into the 2nd equation and solve for [tex]z[/tex]:

[tex]x-6y-7z=-29[/tex]

[tex](-y-z-6)-6y-7=-29[/tex]

[tex]-7y-z-13=-29[/tex]

[tex]-7y-z=-16[/tex]

[tex]-z=-16+7y[/tex]

[tex]z=16-7y[/tex]

Substitute the value of [tex]z[/tex] into the 3rd equation and solve for [tex]y[/tex]:

[tex]-7y-5z=4[/tex]

[tex]-7y-5(16-7y)=4[/tex]

[tex]-7y-80+35y=4[/tex]

[tex]28y-80=4[/tex]

[tex]28y=84[/tex]

[tex]y=3[/tex]

Plug [tex]y=3[/tex] into the solved expression for [tex]z[/tex] and evaluate to solve for [tex]z[/tex]:

[tex]z=16-7(3)[/tex]

[tex]z=16-21[/tex]

[tex]z=-5[/tex]

Plug [tex]z=-5[/tex] into the solved expression for [tex]x[/tex] and evaluate to solve for [tex]x[/tex]:

[tex]x=-(3)-(-5)-6[/tex]

[tex]x=-3+5-6[/tex]

[tex]x=2-6[/tex]

[tex]x=-4[/tex]

Therefore:

[tex]x=-4[/tex]

[tex]y=3[/tex]

[tex]z=-5[/tex]

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