Find the point of intersection of the straight lines :
`(i) 2x+3y-6=0`, `3x-2y-6=0`
`(ii) x=0`, `2x-y+3=0`
`(iii) (x)/(3)-(y)/(4)=0`, `(x)/(2)+(y)/(3)=1`

Answer

To find the points of intersection of the given pairs of straight lines, we will solve each pair of equations step by step.

### (i) Find the point of intersection of the lines:
**Equations:**
1. \( 2x + 3y - 6 = 0 \)
2. \( 3x - 2y - 6 = 0 \)

**Step 1:** Rewrite the equations in standard form.
- From the first equation:
\( 2x + 3y = 6 \)
- From the second equation:
\( 3x - 2y = 6 \)

**Step 2:** Multiply the first equation by 3 and the second equation by 2 to eliminate \(x\).
- First equation: \( 3(2x + 3y) = 3(6) \)
\( 6x + 9y = 18 \)
- Second equation: \( 2(3x - 2y) = 2(6) \)
\( 6x - 4y = 12 \)

**Step 3:** Now, subtract the second equation from the first:
\( (6x + 9y) - (6x - 4y) = 18 - 12 \)
This simplifies to:
\( 13y = 6 \)
So, \( y = \frac{6}{13} \)

**Step 4:** Substitute \( y \) back into one of the original equations to find \( x \).
Using the first equation:
\( 2x + 3\left(\frac{6}{13}\right) = 6 \)
\( 2x + \frac{18}{13} = 6 \)
Multiply through by 13 to eliminate the fraction:
\( 26x + 18 = 78 \)
So, \( 26x = 60 \)
Thus, \( x = \frac{60}{26} = \frac{30}{13} \)

**Final Answer for (i):**
The point of intersection is \( \left(\frac{30}{13}, \frac{6}{13}\right) \).

### (ii) Find the point of intersection of the lines:
**Equations:**
1. \( x = 0 \)
2. \( 2x - y + 3 = 0 \)

**Step 1:** Substitute \( x = 0 \) into the second equation:
\( 2(0) - y + 3 = 0 \)
This simplifies to:
\( -y + 3 = 0 \)
Thus, \( y = 3 \)

**Final Answer for (ii):**
The point of intersection is \( (0, 3) \).

### (iii) Find the point of intersection of the lines:
**Equations:**
1. \( \frac{x}{3} - \frac{y}{4} = 0 \)
2. \( \frac{x}{2} + \frac{y}{3} = 1 \)

**Step 1:** Rewrite the first equation:
\( \frac{x}{3} = \frac{y}{4} \)
Cross-multiplying gives:
\( 4x = 3y \)
Thus, \( y = \frac{4}{3}x \)

**Step 2:** Substitute \( y \) into the second equation:
\( \frac{x}{2} + \frac{4}{3} \cdot \frac{x}{3} = 1 \)
This simplifies to:
\( \frac{x}{2} + \frac{4x}{9} = 1 \)
Finding a common denominator (18):
\( \frac{9x}{18} + \frac{8x}{18} = 1 \)
Thus, \( \frac{17x}{18} = 1 \)
So, \( x = \frac{18}{17} \)

**Step 3:** Substitute \( x \) back to find \( y \):
Using \( y = \frac{4}{3}x \):
\( y = \frac{4}{3} \cdot \frac{18}{17} = \frac{72}{51} = \frac{24}{17} \)

**Final Answer for (iii):**
The point of intersection is \( \left(\frac{18}{17}, \frac{24}{17}\right) \).

### Summary of Answers:
1. \( \left(\frac{30}{13}, \frac{6}{13}\right) \)
2. \( (0, 3) \)
3. \( \left(\frac{18}{17}, \frac{24}{17}\right) \)