If the lines $2x + y - 3 = 0 , 5x + ky - 3 = 0$ and $3x - y - 2 = 0$ are concurrent, find the value of k.

Step 1: Find the intersection of the first two given lines

We are given three lines, and we know they are concurrent, meaning they all intersect at a single point. To find this point, we can solve any two of the given linear equations simultaneously. Let's choose the first and third equations, as they are simpler to solve due to the $y$ terms having opposite signs.

Step 2: Verify the intersection point

The point of intersection of the first and third lines is $(1, 1)$. We can quickly verify this by substituting these coordinates back into both equations to ensure they satisfy both.

Step 3: Substitute the intersection point into the third line equation

Since all three lines are concurrent, the point of intersection $(1, 1)$ must also lie on the second line, $5x + ky - 3 = 0$. We can substitute the coordinates $x=1$ and $y=1$ into this equation to find the value of $k$.

Step 4: Solve for k

Now, we simplify the equation obtained in the previous step to solve for $k$. This gives us the required value of $k$ for which the three lines are concurrent.

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