`{:("Column A"," ","Column B"),("The slope of a line passing through (-3, -4) and the - origin",,"The slope of a line passing through (-5,-6) and the origin"):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to find the slopes of the lines passing through the given points and the origin for both columns. We'll follow the steps outlined below:

### Step-by-Step Solution:

**Step 1: Identify the coordinates for Column A.**
- The points are (-3, -4) and the origin (0, 0).

**Step 2: Use the slope formula for Column A.**
- The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
- Here, let \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (-3, -4)\).

**Step 3: Substitute the values into the slope formula for Column A.**
- Substitute \(y_2 = -4\), \(y_1 = 0\), \(x_2 = -3\), and \(x_1 = 0\):
\[ m_A = \frac{-4 - 0}{-3 - 0} = \frac{-4}{-3} = \frac{4}{3} \]

**Step 4: Identify the coordinates for Column B.**
- The points are (-5, -6) and the origin (0, 0).

**Step 5: Use the slope formula for Column B.**
- Again, using the same slope formula:
- Let \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (-5, -6)\).

**Step 6: Substitute the values into the slope formula for Column B.**
- Substitute \(y_2 = -6\), \(y_1 = 0\), \(x_2 = -5\), and \(x_1 = 0\):
\[ m_B = \frac{-6 - 0}{-5 - 0} = \frac{-6}{-5} = \frac{6}{5} \]

**Step 7: Compare the slopes of Column A and Column B.**
- We have:
- Slope of Column A: \(m_A = \frac{4}{3}\)
- Slope of Column B: \(m_B = \frac{6}{5}\)

**Step 8: Convert the slopes to decimal form for easier comparison.**
- \(m_A = \frac{4}{3} \approx 1.33\)
- \(m_B = \frac{6}{5} = 1.2\)

**Step 9: Determine which slope is larger.**
- Since \(1.33 > 1.2\), we conclude that:
- Column A is greater than Column B.

### Final Conclusion:
- The slope of the line passing through (-3, -4) and the origin is greater than the slope of the line passing through (-5, -6) and the origin.

### Answer:
- Column A is larger than Column B.

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