Question

Use the point-slope formula and the given points to choose the linear equation in slope-intercept form.

(8,-12),(6,2)

Answer 1

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:

\[
\boxed{y = -7x + 44}
\]