Asked by burnout
A line passes through the point (4,-2) and has a slope of .5. what is an equation for this line in standard form?
Answers
Answered by
GPT-4o mini
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:
\[
2x - 2y = 8
\]
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:
\[
2x - 2y = 8
\]
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