To determine the value of $k$ such that the given system of linear equations has no solution, let's first consider the system:

$1) \quad x + ky = 1$ $2) \quad kx + y = 1$

For a system of linear equations to have no solution, the lines must be parallel. This happens when the two equations represent lines with the same slope but different intercepts.

Step 1: Express each equation in slope-intercept form

We will rewrite each equation in the form $y = mx + b$, where $m$ is the slope.

From the first equation $x + ky = 1$:

$ky = -x + 1$ $y = -\frac{1}{k}x + \frac{1}{k}$

So, the slope of the first equation is $m_1 = -\frac{1}{k}$.

From the second equation $kx + y = 1$:

$y = -kx + 1$

The slope of the second equation is $m_2 = -k$.

Step 2: Set the slopes equal for parallel lines

For the lines to be parallel, their slopes must be equal:

$-\frac{1}{k} = -k$

Step 3: Solve for $k$

Solve the equation $-\frac{1}{k} = -k$:

$\frac{1}{k} = k$

Multiply both sides by $k$ to eliminate the denominator:

$1 = k^2$

Thus, $k = \pm 1$.

Step 4: Check the intercepts

For the system to have no solution, the intercepts must be different. We substitute $k = 1$ and $k = -1$ to check:

• For $k = 1$:
  • Equation 1: $x + y = 1$
  • Equation 2: $x + y = 1$

These are the same line, so they have infinitely many solutions.

• For $k = -1$:
  • Equation 1: $x - y = 1$
  • Equation 2: $-x + y = 1$

The slopes are equal, but the equations are different, meaning the lines are parallel with no solution.

Conclusion:

The value of $k$ that makes the system have no solution is:

$\boxed{k = -1}$

Would you like further clarification on any step?

Here are 5 related questions to expand on this:

1. What conditions lead to a system of linear equations having infinite solutions?
2. How do you determine if a system of equations has a unique solution?
3. How can you graphically represent systems of linear equations with no solution?
4. What is the geometric interpretation of parallel lines in linear systems?
5. How do different values of $k$ affect the solution of the system?

Tip: In a system of linear equations, parallel lines always indicate no solutions, while coincident lines indicate infinite solutions.