To find the linear equation in slope-intercept form using the given points \((8, -12)\) and \((6, 2)\), we first need to determine the slope of the line that passes through these points.
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let's assign the points:
- \((x_1, y_1) = (8, -12)\)
- \((x_2, y_2) = (6, 2)\)
Now, substituting the values into the slope formula:
\[ m = \frac{2 - (-12)}{6 - 8} = \frac{2 + 12}{6 - 8} = \frac{14}{-2} = -7 \]
Now that we have the slope \(m = -7\), we can use the point-slope formula to find the equation of the line. The point-slope formula is:
\[ y - y_1 = m(x - x_1) \]
We can use either point for \((x_1, y_1)\). Let's use the point \((8, -12)\):
\[ y - (-12) = -7(x - 8) \]
This simplifies to:
\[ y + 12 = -7(x - 8) \]
Now, distribute the \(-7\):
\[ y + 12 = -7x + 56 \]
Next, isolate \(y\) by subtracting 12 from both sides:
\[ y = -7x + 56 - 12 \]
\[ y = -7x + 44 \]
Now we have the equation in slope-intercept form \(y = mx + b\), where \(m = -7\) and \(b = 44\).
Thus, the final linear equation in slope-intercept form is:
\[ \boxed{y = -7x + 44} \]