The line through the points `(m, -9)` and `(7, m)` has slope m. Then, the x - intercept of this line is

Options

A. `-18`
B. `-6`
C. 6
D. 18

Answer

To find the x-intercept of the line that passes through the points \((m, -9)\) and \((7, m)\) with a slope of \(m\), we can follow these steps:

### Step 1: Calculate the slope of the line
The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, we have the points \((m, -9)\) and \((7, m)\). Thus, we can substitute:
- \(x_1 = m\), \(y_1 = -9\)
- \(x_2 = 7\), \(y_2 = m\)

So, the slope becomes:
\[ m = \frac{m - (-9)}{7 - m} = \frac{m + 9}{7 - m} \]

### Step 2: Set the slope equal to \(m\)
According to the problem, the slope of the line is also \(m\). Therefore, we can set up the equation:
\[ m = \frac{m + 9}{7 - m} \]

### Step 3: Cross-multiply to solve for \(m\)
Cross-multiplying gives us:
\[ m(7 - m) = m + 9 \]
Expanding this, we have:
\[ 7m - m^2 = m + 9 \]

### Step 4: Rearranging the equation
Rearranging the equation leads to:
\[ -m^2 + 7m - m - 9 = 0 \]
This simplifies to:
\[ -m^2 + 6m - 9 = 0 \]
Multiplying through by -1 gives:
\[ m^2 - 6m + 9 = 0 \]

### Step 5: Factor the quadratic equation
The quadratic can be factored as:
\[ (m - 3)^2 = 0 \]
Thus, we find:
\[ m - 3 = 0 \implies m = 3 \]

### Step 6: Substitute \(m\) back to find the line's equation
Now that we have \(m = 3\), we can substitute this back into the points:
- The points are \((3, -9)\) and \((7, 3)\).

### Step 7: Find the equation of the line
Using the point-slope form of the line:
\[ y - y_1 = m(x - x_1) \]
Using point \((3, -9)\):
\[ y - (-9) = 3(x - 3) \]
This simplifies to:
\[ y + 9 = 3x - 9 \implies y = 3x - 18 \]

### Step 8: Find the x-intercept
To find the x-intercept, set \(y = 0\):
\[ 0 = 3x - 18 \]
Solving for \(x\):
\[ 3x = 18 \implies x = 6 \]

### Final Answer
The x-intercept of the line is \(6\).

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