If `(-3, -5)` is the midpoint of the part of the line between the x and y axes, then what is the slope of the line?

Options

A. `-5//3`
B. `-3//5`
C. `3//5`
D. `5//3`

Answer

To find the slope of the line given that the point \((-3, -5)\) is the midpoint of the segment between the x-axis and y-axis, we can follow these steps:

### Step 1: Identify the points on the axes
Let the point where the line intersects the x-axis be \(A(a, 0)\) and the point where it intersects the y-axis be \(B(0, b)\).

### Step 2: Use the midpoint formula
The midpoint \(M\) of the segment \(AB\) can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points \(A\) and \(B\):
\[ M = \left( \frac{a + 0}{2}, \frac{0 + b}{2} \right) = \left( \frac{a}{2}, \frac{b}{2} \right) \]
Given that the midpoint \(M\) is \((-3, -5)\), we can set up the equations:
\[ \frac{a}{2} = -3 \quad \text{and} \quad \frac{b}{2} = -5 \]

### Step 3: Solve for \(a\) and \(b\)
From the first equation:
\[ a = -3 \times 2 = -6 \]
From the second equation:
\[ b = -5 \times 2 = -10 \]
Thus, the coordinates of points \(A\) and \(B\) are \(A(-6, 0)\) and \(B(0, -10)\).

### Step 4: Calculate the slope of the line
The slope \(m\) of the line passing through points \(A\) and \(B\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-10 - 0}{0 - (-6)} = \frac{-10}{6} = -\frac{5}{3} \]

### Final Answer
The slope of the line is:
\[ \boxed{-\frac{5}{3}} \]

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