Solve the system of equations.

Question

left brace Start 3 By 1 Matrix 1st Row 1st Column 2nd Row 1st Column 3rd Row 1st Column EndMatrix

x plus 7 y minus z
equals48
y plus 8 z
equals71
z
equals8
Question content area bottom
Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer boxes to complete the choice.
A.
The solution of the given system is xequals
  
enter your response here​, yequals
  
enter your response here​, and zequals
  
enter your response here.
​(Simplify your​ answers.)
B.
The system has infinitely many solutions. If z is any real​ number, then xequals
  
enter your response here and yequals
  
enter your response here.
​(Type expressions using z as the​ variable.)
C.
The system is inconsistent and has no solution.

Answer 1

To solve the given system of equations, we have:

1. \( x + 7y - z = 48 \) (Equation 1)
2. \( y + 8z = 71 \) (Equation 2)
3. \( z = 8 \) (Equation 3)

Let's substitute \( z = 8 \) (from Equation 3) into the other equations.

Substituting \( z = 8 \) into Equation 2:

\[ y + 8(8) = 71 \] \[ y + 64 = 71 \] \[ y = 71 - 64 \] \[ y = 7 \]

Now substituting \( y = 7 \) and \( z = 8 \) into Equation 1:

\[ x + 7(7) - 8 = 48 \] \[ x + 49 - 8 = 48 \] \[ x + 41 = 48 \] \[ x = 48 - 41 \] \[ x = 7 \]

We have found the values:

• \( x = 7 \)
• \( y = 7 \)
• \( z = 8 \)

Thus, the solution to the system of equations is:

Answer: A. The solution of the given system is \( x = 7 \), \( y = 7 \), and \( z = 8 \).