Question

Which is an equation in point-slope form for the line that passes through the points (−1,4) and (3,−4)?



y−4=2(x+3)
y minus 4 equals 2 open parenthesis x plus 3 close parenthesis

y+4=2(x−3)
y plus 4 equals 2 open parenthesis x minus 3 close parenthesis

y−3=−2(x+4)
y minus 3 equals negative 2 open parenthesis x plus 4 close parenthesis

y+4=−2(x−3)

Answer 1

To find the equation in point-slope form for the line that passes through the points \((-1, 4)\) and \((3, -4)\), we first need to calculate the slope \(m\) of the line using the formula:

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

Here, \((x_1, y_1) = (-1, 4)\) and \((x_2, y_2) = (3, -4)\).

Calculating the slope:

\[
m = \frac{-4 - 4}{3 - (-1)} = \frac{-8}{3 + 1} = \frac{-8}{4} = -2
\]

Now we have the slope \(m = -2\).

Next, we can use the point-slope form of a line, which is:

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

We can use either of the two points, but we’ll use \((-1, 4)\).

Substituting in the values:

\[
y - 4 = -2(x - (-1))
\]
\[
y - 4 = -2(x + 1)
\]

None of the options listed perfectly match this form, but let's transform this equation to see if we can relate it to one of the options.

Working with:
\[
y - 4 = -2(x + 1)
\]

We can expand it:
\[
y - 4 = -2x - 2
\]
\[
y = -2x + 2
\]

Since we were looking for point-slope form and used the point \( (-1, 4) \), it's already in the correct form.

Now let's rewrite the original equation:
We can also compare it to the other options.

**Option:** \(y + 4 = -2(x - 3)\)

Let's check this option:
Using point \((3, -4)\):
\[
y + 4 = -2(x - 3)
\]
- Substitute \(x = 3, y = -4\) into it:
\[
-4 + 4 = -2(3 - 3)
\]
\[
0 = 0 \text{ (which is true, so this option is valid)}
\]

The option \(y + 4 = -2(x - 3)\) is indeed the point-slope form of the equation for the line passing through the specified points.

Thus, the final answer is:

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