The linear equation coincide with the line `2x + 3y = 12 ` is

Options

A. `2x + 3y = 15`
B. ` 7 xx +14y = 13`
C. `8x + 12y = 48`
D. `8x + 10y = 18`

Answer

To find a linear equation that coincides with the line \(2x + 3y = 12\), we need to identify an equation that has the same slope and intercept, which means it must be a scalar multiple of the original equation.

### Step-by-step Solution:

1. **Rewrite the given equation in standard form**:
The given equation is \(2x + 3y = 12\).

2. **Identify coefficients**:
From the equation \(2x + 3y - 12 = 0\), we can identify:
- \(a_1 = 2\)
- \(b_1 = 3\)
- \(c_1 = -12\)

3. **Condition for Coinciding Lines**:
Two lines coincide if the ratios of their coefficients are equal:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

4. **Check each option**:
We will check each option to see if they satisfy the condition for infinite solutions.

**Option 1**: \(2x + 3y - 15 = 0\)
- Here, \(a_2 = 2\), \(b_2 = 3\), \(c_2 = -15\).
- Ratios:
\[ \frac{2}{2} = 1, \quad \frac{3}{3} = 1, \quad \frac{-12}{-15} = \frac{4}{5} \quad \text{(Not equal)} \]
- **Conclusion**: Not coinciding.

**Option 2**: \(7x + 14y - 13 = 0\)
- Here, \(a_2 = 7\), \(b_2 = 14\), \(c_2 = -13\).
- Ratios:
\[ \frac{2}{7}, \quad \frac{3}{14}, \quad \frac{-12}{-13} \quad \text{(Not equal)} \]
- **Conclusion**: Not coinciding.

**Option 3**: \(8x + 12y - 48 = 0\)
- Here, \(a_2 = 8\), \(b_2 = 12\), \(c_2 = -48\).
- Ratios:
\[ \frac{2}{8} = \frac{1}{4}, \quad \frac{3}{12} = \frac{1}{4}, \quad \frac{-12}{-48} = \frac{1}{4} \quad \text{(All equal)} \]
- **Conclusion**: Coinciding.

**Option 4**: \(8x + 10y - 40 = 0\)
- Here, \(a_2 = 8\), \(b_2 = 10\), \(c_2 = -40\).
- Ratios:
\[ \frac{2}{8} = \frac{1}{4}, \quad \frac{3}{10} \quad \text{(Not equal)} \]
- **Conclusion**: Not coinciding.

5. **Final Answer**:
The equation that coincides with the line \(2x + 3y = 12\) is **Option 3: \(8x + 12y - 48 = 0\)**.