https://file.aiquickdraw.com/mathbot/file/1728984764390l20p7z0n.jpg

The problem is asking to find the value of \( k \) such that the linear system:

\[
\begin{cases}
x - 4y = k \\
3x + 2y = 5
\end{cases}
\]

has a solution where \( x = 1 \).

### Step 1: Substitute \( x = 1 \) into both equations.
From the first equation:
\[
x - 4y = k
\]
Substituting \( x = 1 \):
\[
1 - 4y = k \quad \Rightarrow \quad k = 1 - 4y \quad \text{(Equation 1)}
\]

From the second equation:
\[
3x + 2y = 5
\]
Substituting \( x = 1 \):
\[
3(1) + 2y = 5 \quad \Rightarrow \quad 3 + 2y = 5
\]
Solve for \( y \):
\[
2y = 5 - 3 = 2 \quad \Rightarrow \quad y = 1
\]

### Step 2: Substitute \( y = 1 \) into Equation 1.
\[
k = 1 - 4(1) = 1 - 4 = -3
\]

Thus, \( k = -3 \).

### Answer:
The value of \( k \) is \(-3\), so the correct option is **E. -3**.

Would you like further details or have any questions?

### Related Questions:
1. How do you solve a system of equations using substitution in general?
2. What are the other methods for solving systems of equations?
3. How would the solution change if \( x = 2 \) instead of \( x = 1 \)?
4. What is the geometrical interpretation of solving a system of linear equations?
5. Can this system have more than one solution? Why or why not?

### Tip:
When solving systems of equations, substitution is often simpler when one of the variables is already isolated or easily solved for.

The linear system has a solution with x = 1 if k is equal to: 

1. x - 4y = k 
2. 3x + 2y = 5

This problem asks to find the value of k that satisfies a system of linear equations when x = 1. Learn how to solve this system using the substitution method, covering key concepts like linear equations and substitution. Suitable for students in Grades 9-10.

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