In each of these questions two equations numbered I and II are given. You have to solve both the equations and give answer
I `5x^2-87x+378=0`
II `3y^2-49y+200=0`

Options

A. if `xlty`
B. if `xley`
C. if `xgty`
D. if `xgey`

Answer

To solve the given equations step by step, we will start with the first equation and then move on to the second equation.

### Step 1: Solve the first equation \(5x^2 - 87x + 378 = 0\)

1. **Identify the coefficients**:
- Here, \(a = 5\), \(b = -87\), and \(c = 378\).

2. **Calculate the discriminant**:
\[ D = b^2 - 4ac = (-87)^2 - 4 \cdot 5 \cdot 378 \]
\[ D = 7569 - 7560 = 9 \]

3. **Find the roots using the quadratic formula**:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
\[ x = \frac{87 \pm \sqrt{9}}{2 \cdot 5} \]
\[ x = \frac{87 \pm 3}{10} \]

4. **Calculate the two possible values for \(x\)**:
- First root:
\[ x_1 = \frac{87 + 3}{10} = \frac{90}{10} = 9 \]
- Second root:
\[ x_2 = \frac{87 - 3}{10} = \frac{84}{10} = 8.4 \]

### Step 2: Solve the second equation \(3y^2 - 49y + 200 = 0\)

1. **Identify the coefficients**:
- Here, \(a = 3\), \(b = -49\), and \(c = 200\).

2. **Calculate the discriminant**:
\[ D = b^2 - 4ac = (-49)^2 - 4 \cdot 3 \cdot 200 \]
\[ D = 2401 - 2400 = 1 \]

3. **Find the roots using the quadratic formula**:
\[ y = \frac{-b \pm \sqrt{D}}{2a} \]
\[ y = \frac{49 \pm \sqrt{1}}{2 \cdot 3} \]
\[ y = \frac{49 \pm 1}{6} \]

4. **Calculate the two possible values for \(y\)**:
- First root:
\[ y_1 = \frac{49 + 1}{6} = \frac{50}{6} = \frac{25}{3} \approx 8.33 \]
- Second root:
\[ y_2 = \frac{49 - 1}{6} = \frac{48}{6} = 8 \]

### Summary of Solutions:
- The values of \(x\) are \(9\) and \(8.4\).
- The values of \(y\) are approximately \(8.33\) and \(8\).

### Step 3: Compare the values of \(x\) and \(y\)

1. **Comparing \(x\) and \(y\)**:
- For \(x = 9\) and \(y = 8\): \(x > y\)
- For \(x = 9\) and \(y \approx 8.33\): \(x > y\)
- For \(x = 8.4\) and \(y = 8\): \(x > y\)
- For \(x = 8.4\) and \(y \approx 8.33\): \(x > y\)

In all cases, \(x\) is greater than \(y\).

### Final Answer:
- \(x > y\)

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