**Inconsistent System of Linear Equations**

A system of linear equations is said to be inconsistent if it does not have a solution. In other words, there are no values of the variables that satisfy all the equations simultaneously. Let's analyze the system of linear equations given:

1) x + 2y = 3
2) 5x + ky - 7 = 0

To determine whether this system is inconsistent or not, we need to solve it and check if a solution exists.

**Solving the System of Equations:**

We can solve this system of equations by using the method of substitution or elimination. Let's use the method of substitution to find the values of x and y that satisfy both equations:

From equation 1, we have:
x = 3 - 2y

Substituting this value of x into equation 2, we get:
5(3 - 2y) + ky - 7 = 0

Simplifying the equation:
15 - 10y + ky - 7 = 0
8 - 10y + ky = 0

Now, we have a single equation with the variable y. Let's rearrange the equation:
ky - 10y = -8
y(k - 10) = -8

**Analyzing the Coefficient of y:**

For the system to be consistent (have a solution), the coefficient of y must be the same in both equations. However, in this case, the coefficient of y is different in equation 1 and equation 2.

In equation 1: coefficient of y = 2
In equation 2: coefficient of y = k

Since the coefficient of y is not the same, the system is inconsistent if k is not equal to 2.

**Conclusion:**

The system of linear equations x + 2y = 3 and 5x + ky - 7 = 0 is inconsistent if the value of k is not equal to 2. This means that there are no values of x and y that satisfy both equations simultaneously, leading to no solution.