What is an equation of the line that passes through the points left parenthesis, 4, comma, minus, 2, right parenthesis(4,−2) and left parenthesis, 6, comma, 1, right parenthesis(6,1)?

To find the equation of a line passing through two points, we can use the point-slope form:

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

where \( m \) is the slope of the line, and \((x_1, y_1)\) is one of the points on the line. We are given the points \((4, -2)\) and \((6, 1)\).

### Step 1: Find the slope, \( m \)

The slope formula for a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

Substituting the coordinates:

\[
m = \frac{1 - (-2)}{6 - 4} = \frac{1 + 2}{2} = \frac{3}{2}
\]

### Step 2: Use the point-slope form

Now that we have the slope \( m = \frac{3}{2} \), we can substitute it along with one of the points (say, \((4, -2)\)) into the point-slope form:

\[
y - (-2) = \frac{3}{2}(x - 4)
\]

Simplify:

\[
y + 2 = \frac{3}{2}x - 6
\]

### Step 3: Write in slope-intercept form

To convert this to slope-intercept form \( y = mx + b \), subtract 2 from both sides:

\[
y = \frac{3}{2}x - 8
\]

### Final Answer

The equation of the line passing through the points \((4, -2)\) and \((6, 1)\) is:

\[
y = \frac{3}{2}x - 8
\]

Would you like further details, or do you have any questions?

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**Further Practice Questions:**
1. How would you find the equation of a line with a given slope that passes through a specific point?
2. Can you find the equation of a line that passes through the origin and a given point?
3. How do you determine if two lines are parallel or perpendicular using their equations?
4. What would the slope be for a line that is vertical or horizontal?
5. Can you derive the distance formula using points on a line?

**Tip:** Remember, the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) helps determine whether lines are parallel (same slope) or perpendicular (negative reciprocal slopes).

What is an equation of the line that passes through the points (4,−2) and (6,1)?

This solution explains how to find the equation of a line passing through the points (4, -2) and (6, 1). The problem covers key algebra concepts, including slope calculation, point-slope form, and converting to slope-intercept form. Suitable for students in Grades 8-10.

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