Question

-4x - 5y - z = 18
-2x - 5y - 2z = 12
2z + 5y + -2x = 4
what is the solution to this system?
x =
y =
z =
(,, )

Explanation:

Step1: Rewrite the equations

First, rewrite the given system of equations for clarity:

1. \(-4x - 5y - z = 18\)
2. \(-2x - 5y - 2z = 12\)
3. \(-2x + 5y + 2z = 4\) (rewrote \(2z + 5y + -2x = 4\) as \(-2x + 5y + 2z = 4\))

Step2: Eliminate \(y\) from equations 2 and 3

Add equation 2 and equation 3:

$$
LATEXBLOCK0
$$

Step3: Substitute \(x = -4\) into equations 1 and 2

Substitute \(x = -4\) into equation 1: \(-4(-4) - 5y - z = 18 \implies 16 - 5y - z = 18 \implies -5y - z = 2\) (Equation 1a)
Substitute \(x = -4\) into equation 2: \(-2(-4) - 5y - 2z = 12 \implies 8 - 5y - 2z = 12 \implies -5y - 2z = 4\) (Equation 2a)

Step4: Eliminate \(y\) from Equation 1a and Equation 2a

Subtract Equation 1a from Equation 2a:

$$
LATEXBLOCK1
$$

Step5: Substitute \(x = -4\) and \(z = -2\) into Equation 1a

Substitute into \(-5y - z = 2\): \(-5y - (-2) = 2 \implies -5y + 2 = 2 \implies -5y = 0 \implies y = 0\)

Answer:

\(x = -4\), \(y = 0\), \(z = -2\), and the solution to the system is \((-4, 0, -2)\)