Question

This system does not have a unique solution. Solve the system if possible.
left brace Start 2 By 1 Matrix 1st Row 1st Column 2nd Row 1st Column EndMatrix

x minus 8 y plus z equals
9
y minus 3 z equals
8
Question content area bottom
Part 1
Select the correct choice below and fill in any answer boxes within your choice.
A.
There are infinitely many solutions. If z is allowed to be any real​ number, then xequals
  
enter your response here and yequals
  
enter your response here.
​(Type expressions using z as the​ variable.)
B.
There is no solution.

Answer 1

To solve the given system of equations, we can rewrite them in standard form:

1. \( x - 8y + z = 9 \)
2. \( y - 3z = 8 \)

Start with the second equation to express \( y \) in terms of \( z \):

\[
y = 3z + 8
\]

Next, substitute this expression for \( y \) into the first equation:

\[
x - 8(3z + 8) + z = 9
\]

Now simplify the equation:
\[
x - 24z - 64 + z = 9
\]

Combine like terms:
\[
x - 23z - 64 = 9
\]

Now, isolate \( x \):
\[
x - 23z = 9 + 64
\]
\[
x - 23z = 73
\]
\[
x = 73 + 23z
\]

Now we have expressed both \( x \) and \( y \) in terms of \( z \):
- \( x = 73 + 23z \)
- \( y = 3z + 8 \)

Since \( z \) can take any real value, this system has infinitely many solutions.

### Answer:

**A. There are infinitely many solutions. If \( z \) is allowed to be any real number, then:**

\[
x = 73 + 23z \quad \text{and} \quad y = 3z + 8
\]