Question

What is the equation in standard form for a line that passes through the points (4, -6) and (8, 10)?
Responses
A 4x – y = 224 x – y = 22
B 4x – y = -424 x – y = -42
C x – 4y = -32x – 4 y = -32
D x – 4y = -48

Answer 1

To find the equation of a line in standard form that passes through the points \((4, -6)\) and \((8, 10)\), we first need to calculate the slope, \(m\), using the formula:

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

Substituting the points \((x_1, y_1) = (4, -6)\) and \((x_2, y_2) = (8, 10)\):

\[
m = \frac{10 - (-6)}{8 - 4} = \frac{10 + 6}{4} = \frac{16}{4} = 4
\]

Now we have the slope \(m = 4\). Next, we can use the point-slope form of the equation of a line, which is:

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

Using the point \((4, -6)\):

\[
y - (-6) = 4(x - 4)
\]

This simplifies to:

\[
y + 6 = 4x - 16
\]
\[
y = 4x - 16 - 6
\]
\[
y = 4x - 22
\]

Now, to convert this equation into standard form \(Ax + By = C\), we rearrange it:

\[
-4x + y = -22
\]

Multiplying the entire equation by \(-1\) to make \(A\) positive:

\[
4x - y = 22
\]

Thus, in standard form, the equation of the line is:

\[
4x - y = 22
\]

Looking at the provided options, we see that this matches option A:

**A** \(4x - y = 22\)

So the correct answer is:

**A** \(4x - y = 22\)