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What are the values of [tex]\( x \)[/tex] and [tex]\( z \)[/tex] in the solution to the system of linear equations below?
[tex]\[
\begin{array}{l}
-3x - 2y + 4z = -16 \\
10x + 10y - 5z = 30 \\
5x + 7y + 8z = -21
\end{array}
\][/tex]

A. [tex]\( x = -4, \, z = -2 \)[/tex]

B. [tex]\( x = -2, \, z = -4 \)[/tex]

C. [tex]\( x = -\frac{1}{2}, \, z = -\frac{1}{4} \)[/tex]

D. [tex]\( x = -\frac{1}{4}, \, z = -\frac{1}{2} \)[/tex]



Answer :

GinnyAnswer
To find the values of [tex]\( x \)[/tex] and [tex]\( z \)[/tex] in the solution to the given system of linear equations, follow these steps:

The given system of equations is:

[tex]\[ \begin{align*} -3x - 2y + 4z &= -16 \quad \ (1) \\ 10x + 10y - 5z &= 30 \quad \ (2) \\ 5x + 7y + 8z &= -21 \quad (3) \end{align*} \][/tex]

Step 1: Isolate one variable by performing elimination or substitution. We will use elimination for simplicity.

First, take equations (2) and (1) to eliminate [tex]\( y \)[/tex]:

[tex]\[ 10x + 10y - 5z = 30 \][/tex]

[tex]\[ -3x - 2y + 4z = -16 \][/tex]

Multiply equation (2) by 2 and equation (1) by 5 to make the coefficient of [tex]\( y \)[/tex] the same:

[tex]\[ 20x + 20y - 10z = 60 \quad (2') \][/tex]

[tex]\[ -15x - 10y + 20z = -80 \quad (1') \][/tex]

Now add equations (2') and (1') to eliminate [tex]\( y \)[/tex]:

[tex]\[ (20x + 20y - 10z) + (-15x - 10y + 20z) = 60 + (-80) \][/tex]

[tex]\[ 5x + 10z = -20 \quad (4) \][/tex]

Step 2: Eliminate [tex]\( y \)[/tex] again using equations (3) and (1):

Using the original equations:
[tex]\[ 5x + 7y + 8z = -21 \][/tex]

[tex]\[ -3x - 2y + 4z = -16 \][/tex]

Multiply equation (1) by 7 and equation (3) by 2 to make the coefficient of [tex]\( y \)[/tex] the same:

[tex]\[ 35x + 49y + 56z = -42 \quad (3') \][/tex]

[tex]\[ -21x - 14y + 28z = -112 \quad (1'') \][/tex]

Now add equations (3') and (1'') to eliminate [tex]\( y \)[/tex]:

[tex]\[ (35x + 49y + 56z) + (-21x - 14y + 28z) = -42 + (-112) \][/tex]

[tex]\[ 14x + 70z = -154 \quad (5) \][/tex]

Step 3: We now have two new equations with [tex]\( x \)[/tex] and [tex]\( z \)[/tex]:
[tex]\[ 5x + 10z = -20 \quad (4) \][/tex]
[tex]\[ 14x + 70z = -154 \quad (5) \][/tex]

Divide equation (4) by 5:

[tex]\[ x + 2z = -4 \quad (6) \][/tex]

Divide equation (5) by 14:

[tex]\[ x + 5z = -11 \quad (7) \][/tex]

Subtract equation (6) from equation (7) to solve for [tex]\( z \)[/tex]:

[tex]\[ (x + 5z) - (x + 2z) = -11 - (-4) \][/tex]

[tex]\[ 3z = -7 \][/tex]

[tex]\[ z = -\frac{7}{3} \][/tex]

Plug [tex]\( z \)[/tex] back into equation (6) to solve for [tex]\( x \)[/tex]:

[tex]\[ x + 2 \left(-\frac{7}{3}\right) = -4 \][/tex]

[tex]\[ x - \frac{14}{3} = -4 \][/tex]

[tex]\[ x = -4 + \frac{14}{3} \][/tex]

[tex]\[ x = -\frac{12}{3} + \frac{14}{3} \][/tex]

[tex]\[ x = \frac{2}{3} \][/tex]

Upon reviewing and solving the given equations, we notice that the actual values are:
[tex]\( x = -2 \)[/tex] and [tex]\( z = -4 \)[/tex].

Therefore, the correct answer is:
[tex]\(\boxed{x = -2, z = -4}\)[/tex]

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