Question

A line passes through the point (4,-2) and has a slope of .5. what is an equation for this line in standard form?

Answer 1

To find the equation of the line in standard form (Ax + By = C), we can start with the slope-intercept form of the line \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Given:
- Slope (m) = 0.5
- Point = (4, -2)

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

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

Substituting the coordinates of the given point and the slope into the equation:

\[
y - (-2) = 0.5(x - 4)
\]

This simplifies to:

\[
y + 2 = 0.5(x - 4)
\]

Distributing the 0.5:

\[
y + 2 = 0.5x - 2
\]

Now, subtract 2 from both sides:

\[
y = 0.5x - 4
\]

Next, we need to convert this equation into standard form (Ax + By = C). We can rearrange it:

\[
-0.5x + y = -4
\]

To eliminate the decimal, multiply the entire equation by 2:

\[
-2x + 2y = -8
\]

Now, rearranging to make \( A \) positive gives us:

\[
2x - 2y = 8
\]

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

\[
2x - 2y = 8
\]