Question

A line passes through the point (2,9) and has a slope of 1/2. Which option correctly writes the equation of the line in standard form?

Answer 1

To write the equation of a line in standard form, we can start with the point-slope form of the equation of a line, which is given by:

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

where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope.

Given:
- The point \((x_1, y_1) = (2, 9)\)
- The slope \(m = \frac{1}{2}\)

Substituting these values into the point-slope form:

\[
y - 9 = \frac{1}{2}(x - 2)
\]

Now, we can simplify this equation:

1. Distributing the slope on the right side:

\[
y - 9 = \frac{1}{2}x - 1
\]

2. Adding 9 to both sides to isolate \(y\):

\[
y = \frac{1}{2}x + 8
\]

Now, we need to convert this equation into standard form, which is \(Ax + By = C\) where \(A\), \(B\), and \(C\) are integers, and \(A\) should be non-negative.

Starting with:

\[
y = \frac{1}{2}x + 8
\]

We can multiply every term by 2 to eliminate the fraction:

\[
2y = x + 16
\]

Now, rearranging this to standard form:

\[
-x + 2y = 16
\]

To write it in the standard form \(Ax + By = C\) with \(A\) being positive, we multiply the entire equation by -1:

\[
x - 2y = -16
\]

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

\[
x - 2y = -16
\]

So, the final answer in standard form is:

\[
x - 2y = -16
\]