Question

What is the equation in point-slope form of a line that passes through the points (7, −8) and (−4, 6) ?

Responses

y−6=−23(x+4)
y minus 6 equals negative fraction 2 over 3 end fraction open parenthesis x plus 4 close parenthesis

y+6=−23(x−4)
y plus 6 equals negative fraction 2 over 3 end fraction open parenthesis x minus 4 close parenthesis

y−6=−1411(x+4)
y minus 6 equals negative fraction 14 over 11 end fraction open parenthesis x plus 4 close parenthesis

y+6=−1411(x−4)

Answer 1

To find the equation of the line in point-slope form that passes through the points \((7, -8)\) and \((-4, 6)\), we first need to determine the slope \(m\) of the line.

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Substituting in the values from the points \((7, -8)\) and \((-4, 6)\):

\[
m = \frac{6 - (-8)}{-4 - 7} = \frac{6 + 8}{-4 - 7} = \frac{14}{-11} = -\frac{14}{11}
\]

Now that we have the slope \(m = -\frac{14}{11}\), we can use the point-slope form of the equation of a line, which is:

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

We can use either point. Let's use \((7, -8)\):

\[
y - (-8) = -\frac{14}{11}(x - 7)
\]

This simplifies to:

\[
y + 8 = -\frac{14}{11}(x - 7)
\]

Now, let's convert this back to a more familiar point-slope form, or checking available options, we can see if any of the given options can be simplified. If we use the other point \((-4, 6)\):

\[
y - 6 = -\frac{14}{11}(x + 4)
\]

So we have:
\[
y - 6 = -\frac{14}{11}(x + 4)
\]

Thus, the valid equation in point-slope form using the point \((-4, 6)\) is:

\[
\boxed{y - 6 = -\frac{14}{11}(x + 4)}
\]

Among the options given, this corresponds to:

**Response:** \(y - 6 = -\frac{14}{11}(x + 4)\)