Question

solve the system.\

$$\begin{cases}4y + z = -20\\\\-x - 6y - z = 23\\\\-7x - 3y - 6z = 1\\end{cases}$$

enter your answer as an ordered triple (x, y, z).\\((?,\square,\square)\\)

Explanation:

Step1: Eliminate z from first two equations

Add the first equation \(4y + z = -20\) and the second equation \(-x - 6y - z = 23\).
\((4y + z)+(-x - 6y - z)=-20 + 23\)
Simplify: \(-x - 2y=3\), let's call this Equation (4).

Step2: Express z from first equation

From \(4y + z = -20\), we get \(z=-20 - 4y\), call this Equation (5).

Step3: Substitute z into third equation

Substitute \(z = -20 - 4y\) into \(-7x - 3y - 6z = 1\):
\(-7x - 3y - 6(-20 - 4y)=1\)
Expand: \(-7x - 3y + 120 + 24y = 1\)
Simplify: \(-7x + 21y=-119\), divide by -7: \(x - 3y = 17\), call this Equation (6).

Step4: Solve Equation (4) and Equation (6)

Equation (4): \(-x - 2y = 3\)
Equation (6): \(x - 3y = 17\)
Add them: \((-x - 2y)+(x - 3y)=3 + 17\)
Simplify: \(-5y = 20\), so \(y=-4\).

Step5: Find x using y

Substitute \(y = -4\) into Equation (6): \(x - 3(-4)=17\)
\(x + 12 = 17\), so \(x = 5\).

Step6: Find z using y

Substitute \(y = -4\) into Equation (5): \(z=-20 - 4(-4)\)
\(z=-20 + 16=-4\).

Answer:

\((5, -4, -4)\)