Convert the following equations into intercept form and find the intercepts cuts from axes from these lines :
`(i) 4x+3y=24` `(ii) 2x-7y=14`
`(iii) 2x+3y=6` `(iv)3x-y=4`

Answer

To convert the given equations into intercept form and find the intercepts from the axes, we will follow these steps for each equation.

### Step-by-Step Solutions:

**(i) For the equation \(4x + 3y = 24\):**

1. **Convert to intercept form:**
- Divide the entire equation by 24 to make the right side equal to 1.
\[ \frac{4x}{24} + \frac{3y}{24} = 1 \]
- This simplifies to:
\[ \frac{x}{6} + \frac{y}{8} = 1 \]
- Thus, the intercept form is:
\[ \frac{x}{6} + \frac{y}{8} = 1 \]

2. **Find the intercepts:**
- The x-intercept (when \(y = 0\)):
\[ x = 6 \quad \text{(point: (6, 0))} \]
- The y-intercept (when \(x = 0\)):
\[ y = 8 \quad \text{(point: (0, 8))} \]

**Intercepts for (i):**
- x-intercept = 6, y-intercept = 8.

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**(ii) For the equation \(2x - 7y = 14\):**

1. **Convert to intercept form:**
- Divide the entire equation by 14:
\[ \frac{2x}{14} - \frac{7y}{14} = 1 \]
- This simplifies to:
\[ \frac{x}{7} - \frac{y}{2} = 1 \]
- Rearranging gives:
\[ \frac{x}{7} + \frac{y}{-2} = 1 \]

2. **Find the intercepts:**
- The x-intercept (when \(y = 0\)):
\[ x = 7 \quad \text{(point: (7, 0))} \]
- The y-intercept (when \(x = 0\)):
\[ y = -2 \quad \text{(point: (0, -2))} \]

**Intercepts for (ii):**
- x-intercept = 7, y-intercept = -2.

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**(iii) For the equation \(2x + 3y = 6\):**

1. **Convert to intercept form:**
- Divide the entire equation by 6:
\[ \frac{2x}{6} + \frac{3y}{6} = 1 \]
- This simplifies to:
\[ \frac{x}{3} + \frac{y}{2} = 1 \]

2. **Find the intercepts:**
- The x-intercept (when \(y = 0\)):
\[ x = 3 \quad \text{(point: (3, 0))} \]
- The y-intercept (when \(x = 0\)):
\[ y = 2 \quad \text{(point: (0, 2))} \]

**Intercepts for (iii):**
- x-intercept = 3, y-intercept = 2.

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**(iv) For the equation \(3x - y = 4\):**

1. **Convert to intercept form:**
- Divide the entire equation by 4:
\[ \frac{3x}{4} - \frac{y}{4} = 1 \]
- This simplifies to:
\[ \frac{x}{\frac{4}{3}} + \frac{y}{-4} = 1 \]

2. **Find the intercepts:**
- The x-intercept (when \(y = 0\)):
\[ x = \frac{4}{3} \quad \text{(point: }\left(\frac{4}{3}, 0\right)\text{)} \]
- The y-intercept (when \(x = 0\)):
\[ y = -4 \quad \text{(point: (0, -4))} \]

**Intercepts for (iv):**
- x-intercept = \(\frac{4}{3}\), y-intercept = -4.

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### Summary of Intercepts:
1. **(i)** x-intercept = 6, y-intercept = 8.
2. **(ii)** x-intercept = 7, y-intercept = -2.
3. **(iii)** x-intercept = 3, y-intercept = 2.
4. **(iv)** x-intercept = \(\frac{4}{3}\), y-intercept = -4.