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To find the equation in slope-intercept form of the line that passes through the points \((6, -2)\) and \((12, 1)\), let's go through the following steps:

### Step 1: Calculate the Slope (\(m\))
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

For \((x_1, y_1) = (6, -2)\) and \((x_2, y_2) = (12, 1)\):
\[
m = \frac{1 - (-2)}{12 - 6} = \frac{1 + 2}{6} = \frac{3}{6} = \frac{1}{2}
\]

### Step 2: Use the Point-Slope Form
With the slope \(m = \frac{1}{2}\) and a point on the line, say \((6, -2)\), we can use the point-slope form of a line:
\[
y - y_1 = m(x - x_1)
\]

Substitute \(m = \frac{1}{2}\), \(x_1 = 6\), and \(y_1 = -2\):
\[
y - (-2) = \frac{1}{2}(x - 6)
\]
\[
y + 2 = \frac{1}{2}x - 3
\]

### Step 3: Convert to Slope-Intercept Form (\(y = mx + b\))
Now, isolate \(y\) to write the equation in slope-intercept form:
\[
y = \frac{1}{2}x - 3 - 2
\]
\[
y = \frac{1}{2}x - 5
\]

### Final Answer
The equation of the line in slope-intercept form is:
\[
y = \frac{1}{2}x - 5
\]

Would you like further details on this process?

Here are 5 related questions to extend this topic:
1. How do you find the equation of a line that is perpendicular to this line and passes through a given point?
2. What is the equation of a line parallel to this one that passes through a different point?
3. How would you graph this line using only the slope and y-intercept?
4. How can you determine if a third point lies on this line?
5. How do you convert from slope-intercept form to standard form?

**Tip:** In slope-intercept form (\(y = mx + b\)), \(m\) is the slope and \(b\) is the y-intercept, where the line crosses the y-axis.

Write an equation in slope-intercept form of the line that passes through (6, -2) and (12, 1).

Learn how to find the equation of a line in slope-intercept form that passes through points (6, -2) and (12, 1). This tutorial covers key algebra concepts, including calculating slope and using slope-intercept form. Suitable for students in Grades 8-10.

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