Question

solve the following system of equations for x, y, and z using the substitution method. what is the value of z?

$-x + 3y - 6z = 20$

$x + 4y - z = 15$

$-7y - 5z = -23$

(1 point)

$z = \square$

Explanation:

Step1: Solve for \( x \) from the second equation

From \( x + 4y - z = 15 \), we get \( x = 15 - 4y + z \).

Step2: Substitute \( x \) into the first equation

Substitute \( x = 15 - 4y + z \) into \( -x + 3y - 6z = 20 \):

$$ -(15 - 4y + z) + 3y - 6z = 20 $$

Simplify:

$$ -15 + 4y - z + 3y - 6z = 20\\ 7y - 7z = 35\\ y - z = 5\\ y = z + 5 $$

Step3: Substitute \( y = z + 5 \) into the third equation

Substitute \( y = z + 5 \) into \( -7y - 5z = -23 \):

$$ -7(z + 5) - 5z = -23 $$

Simplify:

$$ -7z - 35 - 5z = -23\\ -12z - 35 = -23\\ -12z = 12\\ z = -1 $$

Answer:

\( z = -1 \)