Write an equation, in point-slope form, of the line that passes through the given points: 


( 1 / 2, - 2 ) ( 1, 3 )

To write the equation of the line in point-slope form that passes through the points \(\left(\frac{1}{2}, -2\right)\) and \((1, 3)\), follow these steps:

1. **Find the slope** (\(m\)) of the line using the slope formula:

   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]

   Substituting the coordinates:

   \[
   m = \frac{3 - (-2)}{1 - \frac{1}{2}} = \frac{3 + 2}{\frac{2}{2} - \frac{1}{2}} = \frac{5}{\frac{1}{2}} = 5 \times 2 = 10
   \]

2. **Use the point-slope form equation**:

   \[
   y - y_1 = m(x - x_1)
   \]

   Let's use the point \(\left(\frac{1}{2}, -2\right)\):

   \[
   y - (-2) = 10\left(x - \frac{1}{2}\right)
   \]

3. **Simplify the equation**:

   \[
   y + 2 = 10\left(x - \frac{1}{2}\right)
   \]

This is the equation of the line in point-slope form.

Would you like a more detailed explanation, or do you have any other questions?

Here are 5 related questions for further practice:

1. Write the equation in slope-intercept form for the same line.
2. Find the equation of a line passing through the points \((3, -4)\) and \((6, 2)\).
3. Determine the slope of a line that is parallel to the line passing through \(\left(\frac{1}{2}, -2\right)\) and \((1, 3)\).
4. Write the equation of the line with a slope of \(\frac{3}{2}\) that passes through the point \((2, -1)\).
5. Find the point of intersection between the line passing through \(\left(\frac{1}{2}, -2\right)\) and \((1, 3)\) and the line \(y = -2x + 1\).

**Tip:** When working with fractions in the slope formula, it's often helpful to multiply by the reciprocal to simplify calculations.

Learn how to write the equation of a line in point-slope form passing through the points (1/2, -2) and (1, 3). This problem involves finding the slope using the slope formula and using it to construct the equation in point-slope form. Suitable for Grades 9-12 students.

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