State whether the st.lines :
`x-y-1=0`, `4x+3y=25` and `2x-3y+1=0` are concurrent or not ?

Answer

To determine whether the straight lines given by the equations \(x - y - 1 = 0\), \(4x + 3y - 25 = 0\), and \(2x - 3y + 1 = 0\) are concurrent, we need to find their intersection points and check if they all intersect at a single point.

### Step 1: Solve the first two equations
We will first solve the equations \(x - y - 1 = 0\) and \(4x + 3y - 25 = 0\).

From the first equation, we can express \(y\) in terms of \(x\):
\[ y = x - 1 \tag{1} \]

Now, substitute equation (1) into the second equation:
\[ 4x + 3(x - 1) - 25 = 0 \]
Simplifying this:
\[ 4x + 3x - 3 - 25 = 0 \\ 7x - 28 = 0 \\ 7x = 28 \\ x = 4 \]

### Step 2: Find the value of \(y\)
Now that we have \(x = 4\), we can substitute this value back into equation (1) to find \(y\):
\[ y = 4 - 1 = 3 \]

So, the intersection point of the first two lines is \((4, 3)\).

### Step 3: Check if the intersection point satisfies the third equation
Now we need to check if the point \((4, 3)\) satisfies the third equation \(2x - 3y + 1 = 0\):
\[ 2(4) - 3(3) + 1 = 0 \\ 8 - 9 + 1 = 0 \\ 0 = 0 \]

Since the point \((4, 3)\) satisfies the third equation, all three lines intersect at the same point.

### Conclusion
Thus, the lines \(x - y - 1 = 0\), \(4x + 3y - 25 = 0\), and \(2x - 3y + 1 = 0\) are concurrent.

### Final Answer
Hence, these three lines are concurrent.

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