Question

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

Responses

y+4=−57(x+3)
y plus 4 equals negative fraction 5 over 7 end fraction open parenthesis x plus 3 close parenthesis

y−4=−57(x+3)
y minus 4 equals negative fraction 5 over 7 end fraction open parenthesis x plus 3 close parenthesis

y−3=−57(x+4)
y minus 3 equals negative fraction 5 over 7 end fraction open parenthesis x plus 4 close parenthesis

y+4=57(x+3)

Answer 1

To find the equation of the line in point-slope form, we first need to determine the slope of the line that passes through the points \((4, -1)\) and \((-3, 4)\).

The formula for calculating 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 our points:

\[
m = \frac{4 - (-1)}{-3 - 4} = \frac{4 + 1}{-3 - 4} = \frac{5}{-7} = -\frac{5}{7}
\]

Now that we have the slope, we can use one of the points to write the equation in point-slope form, which is:

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

Using the point \( (4, -1) \), where \( x_1 = 4 \) and \( y_1 = -1 \):

\[
y - (-1) = -\frac{5}{7}(x - 4)
\]

This simplifies to:

\[
y + 1 = -\frac{5}{7}(x - 4)
\]

Now, we can also express it using the other point \((-3, 4)\):

\[
y - 4 = -\frac{5}{7}(x - (-3))
\]

This becomes:

\[
y - 4 = -\frac{5}{7}(x + 3)
\]

Both forms are valid representation.

Now let's compare this with the responses provided:

1. **y + 4 = −\( \frac{5}{7} \)(x + 3)** (Not correct, wrong point)
2. **y − 4 = −\( \frac{5}{7} \)(x + 3)** (This is correct based on point \((-3, 4)\))
3. **y − 3 = −\( \frac{5}{7} \)(x + 4)** (Not correct)
4. **y + 4 = \( \frac{5}{7} \)(x + 3)** (Not correct, wrong slope)

Therefore, the correct answer is:

**y − 4 = −\( \frac{5}{7} \)(x + 3)**.