Which of the following linear equation coincide with the line `4x + 5y = 15`?

Options

A. `8x+ 10y= 25`
B. `2x+ 3y= 7`
C. `7x+ 14y= 17`
D. `12x+ 15y= 45`

Answer

To determine which of the given linear equations coincides with the line represented by the equation \(4x + 5y = 15\), we can use the concept of proportionality of coefficients. Two linear equations coincide if the ratios of their coefficients are equal.

### Step-by-Step Solution:

1. **Identify the coefficients of the given equation**:
The equation is \(4x + 5y = 15\).
- Here, \(A_1 = 4\), \(B_1 = 5\), and \(C_1 = 15\).

2. **Set up the proportionality condition**:
For another equation \(A_2x + B_2y = C_2\) to coincide with the given equation, the following condition must hold:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]

3. **Evaluate each option**:
We will check each option provided to see if they satisfy the proportionality condition.

**Option 1**: \(8x + 10y = 25\)
- Here, \(A_2 = 8\), \(B_2 = 10\), \(C_2 = 25\).
- Check the ratios:
\[ \frac{4}{8} = \frac{1}{2}, \quad \frac{5}{10} = \frac{1}{2}, \quad \frac{15}{25} = \frac{3}{5} \]
- Since \(\frac{15}{25} \neq \frac{1}{2}\), this option does not coincide.

**Option 2**: \(2x + 3y = 7\)
- Here, \(A_2 = 2\), \(B_2 = 3\), \(C_2 = 7\).
- Check the ratios:
\[ \frac{4}{2} = 2, \quad \frac{5}{3} \neq 2, \quad \frac{15}{7} \neq 2 \]
- Since the ratios are not equal, this option does not coincide.

**Option 3**: \(7x + 14y = 17\)
- Here, \(A_2 = 7\), \(B_2 = 14\), \(C_2 = 17\).
- Check the ratios:
\[ \frac{4}{7}, \quad \frac{5}{14} = \frac{5}{14}, \quad \frac{15}{17} \]
- Since the ratios are not equal, this option does not coincide.

**Option 4**: \(12x + 15y = 45\)
- Here, \(A_2 = 12\), \(B_2 = 15\), \(C_2 = 45\).
- Check the ratios:
\[ \frac{4}{12} = \frac{1}{3}, \quad \frac{5}{15} = \frac{1}{3}, \quad \frac{15}{45} = \frac{1}{3} \]
- Since all ratios are equal, this option does coincide.

4. **Conclusion**:
The equation that coincides with \(4x + 5y = 15\) is \(12x + 15y = 45\).