By drawing a graph for each of the equations `3x+y+5=0, 3y-x=5` and `2x+5y=1` on the same graph paper, show that the lines given by these equations are concurrent (i.e. they pass through the same point)
Take 2 cm=1 unit on both the axes.

Answer

To solve the problem of showing that the lines given by the equations \(3x + y + 5 = 0\), \(3y - x = 5\), and \(2x + 5y = 1\) are concurrent, we will follow these steps:

### Step 1: Rewrite the equations in slope-intercept form
1. **Equation 1:** \(3x + y + 5 = 0\)
- Rearranging gives:
\[ y = -3x - 5 \]

2. **Equation 2:** \(3y - x = 5\)
- Rearranging gives:
\[ 3y = x + 5 \implies y = \frac{1}{3}x + \frac{5}{3} \]

3. **Equation 3:** \(2x + 5y = 1\)
- Rearranging gives:
\[ 5y = -2x + 1 \implies y = -\frac{2}{5}x + \frac{1}{5} \]

### Step 2: Create a table of values for each equation
We will choose values of \(x\) to find corresponding \(y\) values for each equation.

**For Equation 1:** \(y = -3x - 5\)
- \(x = 0 \Rightarrow y = -5\) (Point: (0, -5))
- \(x = -1 \Rightarrow y = -3 + 5 = -2\) (Point: (-1, -2))
- \(x = -2 \Rightarrow y = 6 - 5 = 1\) (Point: (-2, 1))

**For Equation 2:** \(y = \frac{1}{3}x + \frac{5}{3}\)
- \(x = 0 \Rightarrow y = \frac{5}{3}\) (Point: (0, 1.67))
- \(x = 3 \Rightarrow y = 1 + \frac{5}{3} = 2.67\) (Point: (3, 2.67))
- \(x = -3 \Rightarrow y = -1 + \frac{5}{3} = \frac{2}{3}\) (Point: (-3, 0.67))

**For Equation 3:** \(y = -\frac{2}{5}x + \frac{1}{5}\)
- \(x = 0 \Rightarrow y = \frac{1}{5}\) (Point: (0, 0.2))
- \(x = 5 \Rightarrow y = -2 + \frac{1}{5} = -1.8\) (Point: (5, -1.8))
- \(x = -5 \Rightarrow y = 2 + \frac{1}{5} = 2.2\) (Point: (-5, 2.2))

### Step 3: Plot the points on graph paper
Using a scale of 2 cm = 1 unit, plot the points obtained from the equations on the same graph paper.

1. For Equation 1, plot the points (0, -5), (-1, -2), and (-2, 1).
2. For Equation 2, plot the points (0, 1.67), (3, 2.67), and (-3, 0.67).
3. For Equation 3, plot the points (0, 0.2), (5, -1.8), and (-5, 2.2).

### Step 4: Draw the lines
Connect the points for each equation with a straight line.

### Step 5: Identify the point of intersection
Observe where all three lines intersect. The point of intersection should be the same for all three lines, confirming that they are concurrent.

### Conclusion
After plotting and drawing the lines, we will find that the three lines intersect at a single point, which confirms that the equations \(3x + y + 5 = 0\), \(3y - x = 5\), and \(2x + 5y = 1\) are concurrent.