x and y are two numbers having opposite signs such that `x^2 : y^2 = 49 : 64`. What is the value of (5x - 6y): (6x - 7y)?

Options

A. `44:65`
B. `13:14`
C. `94:117`
D. `83:98`

Answer

To solve the problem, we need to find the value of \((5x - 6y) : (6x - 7y)\) given that \(x^2 : y^2 = 49 : 64\) and that \(x\) and \(y\) have opposite signs.

### Step-by-Step Solution:

1. **Understanding the Ratio**:
Given \(x^2 : y^2 = 49 : 64\), we can express this as:
\[ \frac{x^2}{y^2} = \frac{49}{64} \]
Taking the square root of both sides, we get:
\[ \frac{x}{y} = \frac{7}{8} \quad \text{or} \quad \frac{x}{y} = -\frac{7}{8} \]
Since \(x\) and \(y\) have opposite signs, we use:
\[ \frac{x}{y} = -\frac{7}{8} \]

2. **Expressing \(x\) in terms of \(y\)**:
From \(\frac{x}{y} = -\frac{7}{8}\), we can express \(x\) as:
\[ x = -\frac{7}{8}y \]

3. **Substituting \(x\) into the expression**:
We need to find the ratio \((5x - 6y) : (6x - 7y)\). Substituting \(x = -\frac{7}{8}y\) into both parts:
\[ 5x - 6y = 5\left(-\frac{7}{8}y\right) - 6y = -\frac{35}{8}y - 6y \]
To combine the terms, convert \(6y\) to a fraction with a denominator of 8:
\[ 6y = \frac{48}{8}y \]
Thus,
\[ 5x - 6y = -\frac{35}{8}y - \frac{48}{8}y = -\frac{83}{8}y \]

4. **Calculating the second part**:
Now, substituting \(x\) into \(6x - 7y\):
\[ 6x - 7y = 6\left(-\frac{7}{8}y\right) - 7y = -\frac{42}{8}y - 7y \]
Again, convert \(7y\) to a fraction with a denominator of 8:
\[ 7y = \frac{56}{8}y \]
Thus,
\[ 6x - 7y = -\frac{42}{8}y - \frac{56}{8}y = -\frac{98}{8}y \]

5. **Finding the ratio**:
Now we can find the ratio:
\[ (5x - 6y) : (6x - 7y) = \left(-\frac{83}{8}y\right) : \left(-\frac{98}{8}y\right) \]
The \(y\) cancels out, and we can simplify:
\[ \frac{83}{98} \]

6. **Final answer**:
Therefore, the value of \((5x - 6y) : (6x - 7y)\) is:
\[ \boxed{83 : 98} \]